A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.
1, 2, 20, 369, 10192, 379850, 17930697, 1027046517, 69216504576, 5363945384274, 469658243947850, 45827641349686636, 4928867833029014503, 579101340954599901152, 73778702335232336908585, 10129059530832922239925140
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..250
Programs
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Maple
seq(sum(binomial(n,k) * abs(combinat[stirling1](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]Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}] -
Maxima
makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);
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PARI
a(n) = sum(k=0, n, binomial(n,k)*abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 2))); \\ Michel Marcus, May 28 2017
Formula
a(n) = Sum_{k=0..n} binomial(n,k) * s(2*k,k) * S(2*n-2*k,n-k).
a(n) ~ m * n^n * c^(2*n) * 2^(3*n-1) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.75643120862616967698..., and m = Sum_{j>=0} StirlingS2(2*j,j) * (2*c-1)^j / (j! * 2^(3*j) * c^(2*j)) = 1.170003674502655133465266152119563086693466... . - Vaclav Kotesovec, May 22 2014