DLMF:26.14.E7 (Q7958)

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DLMF:26.14.E7
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    Statements

    Defining formula
    ⟨ n k ⟩ = ∑ j = 0 n - k ( - 1 ) n - k - j ⁢ j ! ⁢ ( n - j k ) ⁢ S ⁡ ( n , j ) , Eulerian-number 𝑛 𝑘 superscript subscript 𝑗 0 𝑛 𝑘 superscript 1 𝑛 𝑘 𝑗 𝑗 binomial 𝑛 𝑗 𝑘 Stirling-number-second-kind-S 𝑛 𝑗 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)% ^{n-k-j}j!\genfrac{(}{)}{0.0pt}{}{n-j}{k}S\left(n,j\right),}}
    0 references
    DLMF id
    DLMF:26.14.E7
    0 references
    Symbols used
    Q12217
    Defining formula
    ⟨ n k ⟩ Eulerian-number 𝑛 𝑘 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}}}
    xml-id
    C26.S14.SS1.p3.m2addec
    0 references
    Q12131
    Defining formula
    S ⁡ ( n , k ) Stirling-number-second-kind-S 𝑛 𝑘 {\displaystyle{\displaystyle S\left(\NVar{n},\NVar{k}\right)}}
    xml-id
    C26.S8.SS1.p3.m2adec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    Q12176
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1ddec
    0 references
    Q12174
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1edec
    0 references
    Q12172
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1gdec
    0 references
     
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