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

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

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

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);

Mathematica

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}]

Prog

(Maxima) makelist(sum(binomial(n, k)*abs(stirling1(2*k, k))*stirling2(2*n-2*k, n-k), k, 0, n), n, 0, 12);
(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

Crossrefs

Cf. A007820, A187646.

Sequence in context: A060164 A267827 A084948 * A263207 A218306 A009236

Adjacent sequences: A187658 A187659 A187660 * A187662 A187663 A187664

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 12 2011

Status

approved