DLMF:26.14.E12 (Q7963)

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DLMF:26.14.E12
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    Statements

    Defining formula
    S ⁡ ( n , m ) = 1 m ! ⁢ ∑ k = 0 n - 1 ⟨ n k ⟩ ⁢ ( k n - m ) , Stirling-number-second-kind-S 𝑛 𝑚 1 𝑚 superscript subscript 𝑘 0 𝑛 1 Eulerian-number 𝑛 𝑘 binomial 𝑘 𝑛 𝑚 {\displaystyle{\displaystyle S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}% \genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{k}{n-m},}}
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    DLMF id
    DLMF:26.14.E12
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    constraint
    n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}}
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    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.m2aidec
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    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.m2aadec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1acdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
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    Q12174
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1jdec
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    Q12171
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C26.S1.XMD5.m1adec
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    Q12172
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1kdec
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