Stirling Number

Stirling numbers of the first kind

$s(n,k)\backslash ,$

$x^n=\backslash sum\_\{k=1\}^n\; s(n,k)(x)^k$

$(x)^n=x(x+1)(x+2)\backslash cdots(x+n-1).$

$s(n,k)=\backslash leftn\; \backslash \backslash \; k\; \backslash end\{matrix\}\backslash right.$

$(x)\_n\; =\; \backslash sum\_\{k=0\}^n\; s(n,k)\; x^k$

$s(n,k)=(-)^\{k-n\}\; \backslash frac\{\backslash Gamma(n)\}\{\backslash Gamma(k)\}w(n,k-1)$

$w(n,k)=\backslash sum\_\{m=0\}^\{k-1\}\backslash frac\{\backslash Gamma(1-k+m)\}\{\backslash Gamma(1-k)\}H\_\{n-1\}^\{(m+1)\}\; w(n,k-1-m)$

Stirling numbers of the second kind

$B\_n=\backslash sum\_\{k=1\}^n\; S(n,k)$

$(x)\_n=x(x-1)(x-2)\backslash cdots(x-n+1)$

$\backslash sum\_\{k=1\}^n\; S(n,k)(x)\_k=x^n.$

$E(X^n)=\backslash sum\_\{k=1\}^n\; S(n,k)\backslash lambda^k.$

Karamata notation

$s(n,k)=\backslash leftn\; \backslash \backslash \; k\; \backslash end\{matrix\}\backslash right.$

$S(n,k)=\backslash left\backslash \{\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\backslash right\backslash \}.$

See also

External references

• D.E. Knuth, Two notes on notation (TeX source).

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