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

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Crossrefs

Cf. A187646, A384495, A384496.

Programs

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);

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