A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).
1, 2, 24, 516, 16064, 655840, 33157240, 1999679696, 140128848384, 11189643689088, 1003005057594240, 99725721676986240, 10892178742891589792, 1296379044138734510656, 166999512859041432577280, 23149972436862049305233280
Offset: 0
Programs
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Maple
seq(sum(binomial(n,k)*abs(combinat[stirling1](2*k,k))*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);
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Mathematica
Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 15}] -
Maxima
makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);
Formula
a(n) = Sum_{k=0..n} binomial(n,k)*s(2*k,k)*s(2*n-2*k,n-k).
Limit_{n->oo} (a(n)/n!)^(1/n) = -8*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 9.821629929136511797503... - Vaclav Kotesovec, May 30 2025