DLMF:26.8.E33 (Q7861)

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DLMF:26.8.E33
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    Statements

    Defining formula
    S ⁡ ( n , n - k ) = ∑ j = 0 k ( - 1 ) j ⁢ ( n - 1 + j k + j ) ⁢ ( n + k k - j ) ⁢ s ⁡ ( k + j , j ) , Stirling-number-second-kind-S 𝑛 𝑛 𝑘 superscript subscript 𝑗 0 𝑘 superscript 1 𝑗 binomial 𝑛 1 𝑗 𝑘 𝑗 binomial 𝑛 𝑘 𝑘 𝑗 Stirling-number-first-kind-S 𝑘 𝑗 𝑗 {\displaystyle{\displaystyle S\left(n,n-k\right)=\sum_{j=0}^{k}(-1)^{j}% \genfrac{(}{)}{0.0pt}{}{n-1+j}{k+j}\genfrac{(}{)}{0.0pt}{}{n+k}{k-j}\*s\left(k% +j,j\right),}}
    0 references
    DLMF id
    DLMF:26.8.E33
    0 references
    Symbols used
    Q12132
    Defining formula
    s ⁡ ( n , k ) Stirling-number-first-kind-S 𝑛 𝑘 {\displaystyle{\displaystyle s\left(\NVar{n},\NVar{k}\right)}}
    xml-id
    C26.S8.SS1.p1.m5asdec
    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.m2apdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1afdec
    0 references
     
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