Results of Combinatorial Analysis
| DLMF | Formula | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|
| 26.3.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \binom{m}{m-n}} | binomial(m,n)=binomial(m,m - n) |
Binomial[m,n]=Binomial[m,m - n] |
Failure | Successful | Skip | - |
| 26.3.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{m-n} = \frac{m!}{(m-n)!\,n!}} | binomial(m,m - n)=(factorial(m))/(factorial(m - n)*factorial(n)) |
Binomial[m,m - n]=Divide[(m)!,(m - n)!*(n)!] |
Successful | Successful | - | - |
| 26.3.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = 0} | binomial(m,n)= 0 |
Binomial[m,n]= 0 |
Failure | Failure | Skip | Successful |
| 26.3.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{m}\binom{m}{n}x^{n} = (1+x)^{m}} | sum(binomial(m,n)*(x)^(n), n = 0..m)=(1 + x)^(m) |
Sum[Binomial[m,n]*(x)^(n), {n, 0, m}]=(1 + x)^(m) |
Successful | Successful | - | - |
| 26.3.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{m=0}^{\infty}\binom{m+n}{m}x^{m} = \frac{1}{(1-x)^{n+1}}} | sum(binomial(m + n,m)*(x)^(m), m = 0..infinity)=(1)/((1 - x)^(n + 1)) |
Sum[Binomial[m + n,m]*(x)^(m), {m, 0, Infinity}]=Divide[1,(1 - x)^(n + 1)] |
Successful | Failure | - | Successful |
| 26.3.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \binom{m-1}{n}+\binom{m-1}{n-1}} | binomial(m,n)=binomial(m - 1,n)+binomial(m - 1,n - 1) |
Binomial[m,n]=Binomial[m - 1,n]+Binomial[m - 1,n - 1] |
Successful | Successful | - | - |
| 26.3.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \frac{m}{n}\binom{m-1}{n-1}} | binomial(m,n)=(m)/(n)*binomial(m - 1,n - 1) |
Binomial[m,n]=Divide[m,n]*Binomial[m - 1,n - 1] |
Successful | Successful | - | - |
| 26.3.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{m}{n}\binom{m-1}{n-1} = \frac{m-n+1}{n}\binom{m}{n-1}} | (m)/(n)*binomial(m - 1,n - 1)=(m - n + 1)/(n)*binomial(m,n - 1) |
Divide[m,n]*Binomial[m - 1,n - 1]=Divide[m - n + 1,n]*Binomial[m,n - 1] |
Successful | Successful | - | - |
| 26.3.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m+1}{n+1} = \sum_{k=n}^{m}\binom{k}{n}} | binomial(m + 1,n + 1)= sum(binomial(k,n), k = n..m) |
Binomial[m + 1,n + 1]= Sum[Binomial[k,n], {k, n, m}] |
Successful | Successful | - | - |
| 26.3.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}} | binomial(m,n)= sum(binomial(m - n - 1 + k,k), k = 0..n) |
Binomial[m,n]= Sum[Binomial[m - n - 1 + k,k], {k, 0, n}] |
Successful | Successful | - | - |
| 26.3.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{n}{0} = \binom{n}{n}} | binomial(n,0)=binomial(n,n) |
Binomial[n,0]=Binomial[n,n] |
Successful | Successful | - | - |
| 26.3.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{n}{n} = 1} | binomial(n,n)= 1 |
Binomial[n,n]= 1 |
Successful | Successful | - | - |
| 26.3.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \sum_{k=0}^{n}(-1)^{n-k}\binom{m+1}{k}} | binomial(m,n)= sum((- 1)^(n - k)*binomial(m + 1,k), k = 0..n) |
Binomial[m,n]= Sum[(- 1)^(n - k)*Binomial[m + 1,k], {k, 0, n}] |
Successful | Failure | - | Successful |
| 26.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \multinomial{n_{1}+n_{2}}{n_{1},n_{2}} = \binom{n_{1}+n_{2}}{n_{1}}} | multinomial(n[1]+ n[2], n[1], n[2])=binomial(n[1]+ n[2],n[1]) |
Multinomial[Subscript[n, 1]+ Subscript[n, 2]]=Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]] |
Failure | Failure | Error | Fail
Complex[1.1103428718503567, -2.707788083134227] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-6.999993813234755, 0.0] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[6.885584583718428, -3.3383853150444147] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.2573870593198788, -0.21521656299188482] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{n_{1}+n_{2}}{n_{1}} = \binom{n_{1}+n_{2}}{n_{2}}} | binomial(n[1]+ n[2],n[1])=binomial(n[1]+ n[2],n[2]) |
Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]]=Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 2]] |
Failure | Successful | Successful | - |
| 26.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{n+1}\binom{2n}{n} = \frac{1}{2n+1}\binom{2n+1}{n}} | (1)/(n + 1)*binomial(2*n,n)=(1)/(2*n + 1)*binomial(2*n + 1,n) |
Divide[1,n + 1]*Binomial[2*n,n]=Divide[1,2*n + 1]*Binomial[2*n + 1,n] |
Successful | Successful | - | - |
| 26.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{2n+1}\binom{2n+1}{n} = \binom{2n}{n}-\binom{2n}{n-1}} | (1)/(2*n + 1)*binomial(2*n + 1,n)=binomial(2*n,n)-binomial(2*n,n - 1) |
Divide[1,2*n + 1]*Binomial[2*n + 1,n]=Binomial[2*n,n]-Binomial[2*n,n - 1] |
Successful | Failure | - | Successful |
| 26.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{2n}{n}-\binom{2n}{n-1} = \binom{2n-1}{n}-\binom{2n-1}{n+1}} | binomial(2*n,n)-binomial(2*n,n - 1)=binomial(2*n - 1,n)-binomial(2*n - 1,n + 1) |
Binomial[2*n,n]-Binomial[2*n,n - 1]=Binomial[2*n - 1,n]-Binomial[2*n - 1,n + 1] |
Failure | Failure | Skip | Successful |
| 26.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{0} = 1} | BellB(0, 1)= 1 |
BellB[0]= 1 |
Successful | Successful | - | - |
| 26.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k}} | BellB(n, 1)= sum(Stirling2(n, k), k = 0..n) |
BellB[n]= Sum[StirlingS2[n, k], {k, 0, n}] |
Failure | Successful | Skip | - |
| 26.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \sum_{k=1}^{m}\frac{k^{n}}{k!}\sum_{j=0}^{m-k}\frac{(-1)^{j}}{j!}} | BellB(n, 1)= sum(((k)^(n))/(factorial(k))*sum(((- 1)^(j))/(factorial(j)), j = 0..m - k), k = 1..m) |
BellB[n]= Sum[Divide[(k)^(n),(k)!]*Sum[Divide[(- 1)^(j),(j)!], {j, 0, m - k}], {k, 1, m}] |
Failure | Failure | Skip | Successful |
| 26.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!}} | BellB(n, 1)= exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity) |
BellB[n]= Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}] |
Failure | Failure | Skip | Fail
Complex[0.8939534673502062, 0.8939534673502062] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.8939534673502062, 1.934473657395984] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[1.934473657395984, 1.934473657395984] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.934473657395984, 0.8939534673502062] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}}} | exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity)= 1 + floor(exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..2*n)) |
Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}]= 1 + Floor[Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, 2*n}]] |
Failure | Failure | Skip | Fail
Complex[-0.47973990497711105, 0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.47973990497711105, -0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.520260095022889, -0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.520260095022889, 0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.7.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{x}-1)} | sum(BellB(n, 1)*((x)^(n))/(factorial(n)), n = 0..infinity)= exp(exp(x)- 1) |
Sum[BellB[n]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]= Exp[Exp[x]- 1] |
Failure | Successful | Skip | - |
| 26.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k}} | BellB(n + 1, 1)= sum(binomial(n,k)*BellB(k, 1), k = 0..n) |
BellB[n + 1]= Sum[Binomial[n,k]*BellB[k], {k, 0, n}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.7#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n}} | BellB(n + 1, 1)= sum(binomial(n,k)*BellB(n, 1), k = 0..n) |
BellB[n + 1]= Sum[Binomial[n,k]*BellB[n], {k, 0, n}] |
Failure | Failure | Skip | Fail
-3.0 <- {Rule[n, 2]}-25.0 <- {Rule[n, 3]} |
| 26.7.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N\ln@@{N} = n} | N*ln(N)= n |
N*Log[N]= n |
Failure | Failure | Fail -1.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 1}-2.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 2}-3.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 3}-1.130462591-2.090978877*I <- {N = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Fail
Complex[-1.1304625910710442, 2.090978878008139] <- {Rule[n, 1], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-2.130462591071044, 2.090978878008139] <- {Rule[n, 2], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-3.130462591071044, 2.090978878008139] <- {Rule[n, 3], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-1.1304625910710442, -2.090978878008139] <- {Rule[n, 1], Rule[N, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n}{n} = 1} | Stirling1(n, n)= 1 |
StirlingS1[n, n]= 1 |
Successful | Failure | - | Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{1}{k} = \Kroneckerdelta{1}{k}} | Stirling1(1, k)= KroneckerDelta[1, k] |
StirlingS1[1, k]= KroneckerDelta[1, k] |
Failure | Failure | Fail -.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 1}-1.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 2}-1.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 3}-.414213562+1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)-I*2^(1/2), k = 1}... skip entries to safe data |
Successful |
| 26.8.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{n} = 1} | Stirling2(n, n)= 1 |
StirlingS2[n, n]= 1 |
Successful | Failure | - | Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{k} = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^{n}} | Stirling2(n, k)=(1)/(factorial(k))*sum((- 1)^(k - j)*binomial(k,j)*(j)^(n), j = 0..k) |
StirlingS2[n, k]=Divide[1,(k)!]*Sum[(- 1)^(k - j)*Binomial[k,j]*(j)^(n), {j, 0, k}] |
Failure | Failure | Skip | Successful |
| 26.8.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}\Stirlingnumbers@{n}{k}x^{k} = (x-n+1)_{n}} | sum(Stirling1(n, k)*(x)^(k), k = 0..n)=x - n + 1[n] |
Sum[StirlingS1[n, k]*(x)^(k), {k, 0, n}]=Subscript[x - n + 1, n] |
Failure | Failure | Skip | Successful |
| 26.8.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!} = \frac{(\ln@{1+x})^{k}}{k!}} | sum(Stirling1(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity)=((ln(1 + x))^(k))/(factorial(k)) |
Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]=Divide[(Log[1 + x])^(k),(k)!] |
Failure | Failure | Skip | Skip |
| 26.8.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n,k=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!}y^{k} = (1+x)^{y}} | sum(sum(Stirling1(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity)=(1 + x)^(y) |
Sum[Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}], {n, 0, Infinity}]=(1 + x)^(y) |
Failure | Failure | Skip | Skip |
| 26.8.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{n}\StirlingnumberS@{n}{k}(x-k+1)_{k} = x^{n}} | sum(Stirling2(n, k)*x - k + 1[k], k = 1..n)= (x)^(n) |
Sum[StirlingS2[n, k]*Subscript[x - k + 1, k], {k, 1, n}]= (x)^(n) |
Failure | Failure | Skip | Successful |
| 26.8.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!} = \frac{(\expe^{x}-1)^{k}}{k!}} | sum(Stirling2(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity)=((exp(x)- 1)^(k))/(factorial(k)) |
Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]=Divide[(Exp[x]- 1)^(k),(k)!] |
Failure | Failure | Skip | Skip |
| 26.8.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n,k=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!}y^{k} = \exp\left(y(\expe^{x}-1)\right)} | sum(sum(Stirling2(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity)= exp(y*(exp(x)- 1)) |
Sum[Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}], {n, 0, Infinity}]= Exp[y*(Exp[x]- 1)] |
Failure | Failure | Skip | Skip |
| 26.8#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n}{0} = 0} | Stirling1(n, 0)= 0 |
StirlingS1[n, 0]= 0 |
Failure | Failure | Successful | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n}{1} = (-1)^{n-1}(n-1)!} | Stirling1(n, 1)=(- 1)^(n - 1)*factorial(n - 1) |
StirlingS1[n, 1]=(- 1)^(n - 1)*(n - 1)! |
Failure | Failure | Successful | Successful |
| 26.8.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\Stirlingnumbers@{n}{n-1} = \StirlingnumberS@{n}{n-1}} | - Stirling1(n, n - 1)= Stirling2(n, n - 1) |
- StirlingS1[n, n - 1]= StirlingS2[n, n - 1] |
Successful | Failure | - | Fail
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}-2.8284271247461903 <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}-2.8284271247461903 <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{n-1} = \binom{n}{2}} | Stirling2(n, n - 1)=binomial(n,2) |
StirlingS2[n, n - 1]=Binomial[n,2] |
Successful | Failure | - | Successful |
| 26.8#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{0} = 0} | Stirling2(n, 0)= 0 |
StirlingS2[n, 0]= 0 |
Failure | Failure | Successful | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{1} = 1} | Stirling2(n, 1)= 1 |
StirlingS2[n, 1]= 1 |
Failure | Failure | Successful | Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{2} = 2^{n-1}-1} | Stirling2(n, 2)= (2)^(n - 1)- 1 |
StirlingS2[n, 2]= (2)^(n - 1)- 1 |
Failure | Failure | Successful | Successful |
| 26.8.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n}{k} = \Stirlingnumbers@{n-1}{k-1}-(n-1)\Stirlingnumbers@{n-1}{k}} | Stirling1(n, k)= Stirling1(n - 1, k - 1)-(n - 1)* Stirling1(n - 1, k) |
StirlingS1[n, k]= StirlingS1[n - 1, k - 1]-(n - 1)* StirlingS1[n - 1, k] |
Failure | Failure | Successful | Successful |
| 26.8.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{k}{h}\Stirlingnumbers@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\Stirlingnumbers@{n-j}{h}\Stirlingnumbers@{j}{k-h}} | binomial(k,h)*Stirling1(n, k)= sum(binomial(n,j)*Stirling1(n - j, h)*Stirling1(j, k - h), j = k - h..n - h) |
Binomial[k,h]*StirlingS1[n, k]= Sum[Binomial[n,j]*StirlingS1[n - j, h]*StirlingS1[j, k - h], {j, k - h, n - h}] |
Failure | Failure | Skip | Successful |
| 26.8.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n+1}{k+1} = n!\sum_{j=k}^{n}\frac{(-1)^{n-j}}{j!}\,\Stirlingnumbers@{j}{k}} | Stirling1(n + 1, k + 1)= factorial(n)*sum(((- 1)^(n - j))/(factorial(j))*Stirling1(j, k), j = k..n) |
StirlingS1[n + 1, k + 1]= (n)!*Sum[Divide[(- 1)^(n - j),(j)!]*StirlingS1[j, k], {j, k, n}] |
Failure | Failure | Skip | Successful |
| 26.8.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n+k+1}{k} = -\sum_{j=0}^{k}(n+j)\Stirlingnumbers@{n+j}{j}} | Stirling1(n + k + 1, k)= - sum((n + j)* Stirling1(n + j, j), j = 0..k) |
StirlingS1[n + k + 1, k]= - Sum[(n + j)* StirlingS1[n + j, j], {j, 0, k}] |
Failure | Failure | Skip | Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{k} = k\StirlingnumberS@{n-1}{k}+\StirlingnumberS@{n-1}{k-1}} | Stirling2(n, k)= k*Stirling2(n - 1, k)+ Stirling2(n - 1, k - 1) |
StirlingS2[n, k]= k*StirlingS2[n - 1, k]+ StirlingS2[n - 1, k - 1] |
Failure | Failure | Successful | Successful |
| 26.8.E23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{k}{h}\StirlingnumberS@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\StirlingnumberS@{n-j}{h}\StirlingnumberS@{j}{k-h}} | binomial(k,h)*Stirling2(n, k)= sum(binomial(n,j)*Stirling2(n - j, h)*Stirling2(j, k - h), j = k - h..n - h) |
Binomial[k,h]*StirlingS2[n, k]= Sum[Binomial[n,j]*StirlingS2[n - j, h]*StirlingS2[j, k - h], {j, k - h, n - h}] |
Failure | Failure | Skip | Successful |
| 26.8.E24 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{k} = \sum_{j=k}^{n}\StirlingnumberS@{j-1}{k-1}k^{n-j}} | Stirling2(n, k)= sum(Stirling2(j - 1, k - 1)*(k)^(n - j), j = k..n) |
StirlingS2[n, k]= Sum[StirlingS2[j - 1, k - 1]*(k)^(n - j), {j, k, n}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n+1}{k+1} = \sum_{j=k}^{n}\binom{n}{j}\StirlingnumberS@{j}{k}} | Stirling2(n + 1, k + 1)= sum(binomial(n,j)*Stirling2(j, k), j = k..n) |
StirlingS2[n + 1, k + 1]= Sum[Binomial[n,j]*StirlingS2[j, k], {j, k, n}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n+k+1}{k} = \sum_{j=0}^{k}j\StirlingnumberS@{n+j}{j}} | Stirling2(n + k + 1, k)= sum(j*Stirling2(n + j, j), j = 0..k) |
StirlingS2[n + k + 1, k]= Sum[j*StirlingS2[n + j, j], {j, 0, k}] |
Failure | Successful | Skip | - |
| 26.8.E27 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Stirlingnumbers@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\,\binom{n+k}{k-j}\*\StirlingnumberS@{k+j}{j}} | Stirling1(n, n - k)= sum((- 1)^(j)*binomial(n - 1 + j,k + j)*binomial(n + k,k - j)* Stirling2(k + j, j), j = 0..k) |
StirlingS1[n, n - k]= Sum[(- 1)^(j)*Binomial[n - 1 + j,k + j]*Binomial[n + k,k - j]* StirlingS2[k + j, j], {j, 0, k}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{n}\Stirlingnumbers@{n}{k} = 0} | sum(Stirling1(n, k), k = 1..n)= 0 |
Sum[StirlingS1[n, k], {k, 1, n}]= 0 |
Failure | Failure | Skip | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 26.8.E29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{n}(-1)^{n-k}\Stirlingnumbers@{n}{k} = n!} | sum((- 1)^(n - k)* Stirling1(n, k), k = 1..n)= factorial(n) |
Sum[(- 1)^(n - k)* StirlingS1[n, k], {k, 1, n}]= (n)! |
Failure | Failure | Skip | Fail
-2.0 <- {Rule[n, 2]}-7.0 <- {Rule[n, 3]} |
| 26.8.E30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=k}^{n}\Stirlingnumbers@{n+1}{j+1}\,n^{j-k} = \Stirlingnumbers@{n}{k}} | sum(Stirling1(n + 1, j + 1)*(n)^(j - k), j = k..n)= Stirling1(n, k) |
Sum[StirlingS1[n + 1, j + 1]*(n)^(j - k), {j, k, n}]= StirlingS1[n, k] |
Failure | Successful | Skip | - |
| 26.8.E33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\binom{n+k}{k-j}\*\Stirlingnumbers@{k+j}{j}} | Stirling2(n, n - k)= sum((- 1)^(j)*binomial(n - 1 + j,k + j)*binomial(n + k,k - j)* Stirling1(k + j, j), j = 0..k) |
StirlingS2[n, n - k]= Sum[(- 1)^(j)*Binomial[n - 1 + j,k + j]*Binomial[n + k,k - j]* StirlingS1[k + j, j], {j, 0, k}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E34 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=0}^{n}j^{k}x^{j} = \sum_{j=0}^{k}\StirlingnumberS@{k}{j}x^{j}\deriv[j]{}{x}\left(\frac{1-x^{n+1}}{1-x}\right)} | sum((j)^(k)* (x)^(j), j = 0..n)= sum(Stirling2(k, j)*(x)^(j)* diff((1 - (x)^(n + 1))/(1 - x), [x$(j)]), j = 0..k) |
Sum[(j)^(k)* (x)^(j), {j, 0, n}]= Sum[StirlingS2[k, j]*(x)^(j)* D[Divide[1 - (x)^(n + 1),1 - x], {x, j}], {j, 0, k}] |
Failure | Failure | Skip | Successful |
| 26.8.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=0}^{n}j^{k} = \sum_{j=0}^{k}j!\StirlingnumberS@{k}{j}\binom{n+1}{j+1}} | sum((j)^(k), j = 0..n)= sum(factorial(j)*Stirling2(k, j)*binomial(n + 1,j + 1), j = 0..k) |
Sum[(j)^(k), {j, 0, n}]= Sum[(j)!*StirlingS2[k, j]*Binomial[n + 1,j + 1], {j, 0, k}] |
Failure | Failure | Skip | Successful |
| 26.8.E36 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}(-1)^{n-k}k!\StirlingnumberS@{n}{k} = 1} | sum((- 1)^(n - k)* factorial(k)*Stirling2(n, k), k = 0..n)= 1 |
Sum[(- 1)^(n - k)* (k)!*StirlingS2[n, k], {k, 0, n}]= 1 |
Failure | Failure | Skip | Successful |
| 26.8.E39 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=k}^{n}\Stirlingnumbers@{j}{k}\StirlingnumberS@{n}{j} = \sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k}} | sum(Stirling1(j, k)*Stirling2(n, j), j = k..n)= sum(Stirling1(n, j)*Stirling2(j, k), j = k..n) |
Sum[StirlingS1[j, k]*StirlingS2[n, j], {j, k, n}]= Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}] |
Failure | Failure | Skip | Fail
Complex[0.0, -2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}-2.8284271247461903 <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, 2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 26.8.E39 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k} = \Kroneckerdelta{n}{k}} | sum(Stirling1(n, j)*Stirling2(j, k), j = k..n)= KroneckerDelta[n, k] |
Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}]= KroneckerDelta[n, k] |
Failure | Failure | Skip | Successful |
| 26.10.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \prod_{j=1}^{\infty}(1+q^{j}) = \prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}} | product(1 + (q)^(j), j = 1..infinity)= product((1)/(1 - (q)^(2*j - 1)), j = 1..infinity) |
Product[1 + (q)^(j), {j, 1, Infinity}]= Product[Divide[1,1 - (q)^(2*j - 1)], {j, 1, Infinity}] |
Failure | Failure | Skip | Successful |
| 26.12.E23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h<j\leq r}\frac{1-q^{3(h+2j-1)}}{1-q^{3(h+j-1)}} = \prod_{h=1}^{r}\left(\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{j=h}^{r}\frac{1-q^{3(r+h+j-1)}}{1-q^{3(2h+j-1)}}\right)} | product((1 - (q)^(3*h - 1))/(1 - (q)^(3*h - 2)), h = 1..r)*product(product((1 - (q)^(3*(h + 2*j - 1)))/(1 - (q)^(3*(h + j - 1))), j = h + 1..r), h = 1..j - 1)= product((1 - (q)^(3*h - 1))/(1 - (q)^(3*h - 2))*product((1 - (q)^(3*(r + h + j - 1)))/(1 - (q)^(3*(2*h + j - 1))), j = h..r), h = 1..r) |
Product[Divide[1 - (q)^(3*h - 1),1 - (q)^(3*h - 2)], {h, 1, r}]*Product[Product[Divide[1 - (q)^(3*(h + 2*j - 1)),1 - (q)^(3*(h + j - 1))], {j, h + 1, r}], {h, 1, j - 1}]= Product[Divide[1 - (q)^(3*h - 1),1 - (q)^(3*h - 2)]*Product[Divide[1 - (q)^(3*(r + h + j - 1)),1 - (q)^(3*(2*h + j - 1))], {j, h, r}], {h, 1, r}] |
Failure | Failure | Skip | Error |
| 26.12#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{3} = 1.20205\;69032} | Zeta(3)= 1.2020569032 |
Zeta[3]= 1.2020569032 |
Successful | Failure | - | Successful |
| 26.12#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta'@{-1} = -0.16542\;11437} | subs( temp=- 1, diff( Zeta(temp), temp$(1) ) )= - 0.1654211437 |
(D[Zeta[temp], {temp, 1}]/.temp-> - 1)= - 0.1654211437 |
Successful | Failure | - | Successful |
| 26.13.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d(n) = n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!}} | d*(n)= factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n) |
d*(n)= (n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}] |
Failure | Failure | Skip | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1]}Complex[1.8284271247461903, 2.8284271247461903] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2]}Complex[2.2426406871192857, 4.242640687119286] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[d, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1]}... skip entries to safe data |
| 26.13.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!} = \floor{\frac{n!+\expe-2}{\expe}}} | factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n)= floor((factorial(n)+ exp(1)- 2)/(exp(1))) |
(n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}]= Floor[Divide[(n)!+ E - 2,E]] |
Failure | Failure | Skip | Successful |
| 26.15.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}} | r[k]*(B)=(2*n)/(2*n - k)*binomial(2*n - k,k) |
Subscript[r, k]*(B)=Divide[2*n,2*n - k]*Binomial[2*n - k,k] |
Failure | Failure | Fail -2.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 1}-4.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 2}-6.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 3}Float(-infinity)+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 2, n = 1}... skip entries to safe data |
Successful |
| 26.15.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2(n!)N_{0}(B) = 2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}\binom{2n-k}{k}{(n-k)!}} | 2*(factorial(n))* N[0]*(B)= 2*(factorial(n))* sum((- 1)^(k)*(2*n)/(2*n - k)*binomial(2*n - k,k)*factorial(n - k), k = 0..n) |
2*((n)!)* Subscript[N, 0]*(B)= 2*((n)!)* Sum[(- 1)^(k)*Divide[2*n,2*n - k]*Binomial[2*n - k,k]*(n - k)!, {k, 0, n}] |
Failure | Failure | Skip | Successful |
| 26.15.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r_{n-k}(B) = \StirlingnumberS@{n}{k}} | r[n - k]*(B)= Stirling2(n, k) |
Subscript[r, n - k]*(B)= StirlingS2[n, k] |
Failure | Failure | Fail -1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 1}-1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 2}-1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 3}0.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 2, n = 1}... skip entries to safe data |
Fail
Complex[-1.4142135623730951, 2.585786437626905] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2.585786437626905, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -5.414213562373095] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-5.414213562373095, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |