A187657 Binomial convolution of the central Stirling numbers of the second kind.
1, 2, 16, 222, 4416, 114660, 3676814, 140408338, 6222858240, 314006546124, 17774855765140, 1115507717954432, 76871991664546170, 5770732305836768712, 468750121409142448386, 40964179307489016777630, 3832326196169482368117760
Offset: 0
Programs
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Maple
seq(sum(binomial(n, k) *combinat[stirling2](2*k, k) *combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);
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Mathematica
Table[Sum[Binomial[n, k]StirlingS2[2k, k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 16}] -
Maxima
makelist(sum(binomial(n,k)*stirling2(2*k,k)*stirling2(2*n-2*k, n-k),k,0,n),n,0,12);
Formula
a(n) = Sum_{k=0..n} binomial(n,k) * S(2k,k) * S(2n-2k,n-k).
Limit n->infinity (a(n)/n!)^(1/n) = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348... . - Vaclav Kotesovec, Jun 01 2015