A187655 Self-convolution of the central Stirling numbers of the second kind.

1, 2, 15, 194, 3631, 89712, 2764268, 102207394, 4411265695, 217707856946, 12092696127691, 746552539553152, 50708165735187572, 3757864633323765824, 301719332111553586612, 26089939284112306045362, 2417245528055399202851119

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Comments

The sequence of the central Stirling numbers of the second kind is 1, 1, 7, 90, 1701,... with offset 0 (see A007820).

Crossrefs

Cf. A187656.

Programs

Maple

seq( add(combinat[stirling2](2*k,k) *combinat[stirling2](2*(n-k),n-k) ,k=0..n), n=0..12);

Mathematica

Table[Sum[StirlingS2[2k, k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 16}]

Maxima

makelist(sum(stirling2(2*k,k)*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

Formula

a(n) = sum_{k=0..n} A048993(2k,k)*A048993(2n-2k,n-k).

a(n) ~ 2^(2*n+1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * (c*(2-c))^n), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014