A187651 Alternated binomial partial sums of the central Stirling numbers of the second kind.

1, 0, 6, 71, 1380, 34854, 1092317, 40900215, 1781924888, 88569337730, 4946558473226, 306691008191732, 20903038895529727, 1553426761730508586, 125016067017985968931, 10831572432055401760624, 1005245087722396707881648

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Crossrefs

Cf. A187653.

Programs

Maple

seq(add((-1)^(n-k)*binomial(n,k)*combinat[stirling2](2*k,k), k=0..n), n=0..20);

Mathematica

Table[Sum[(-1)^(n-k)Binomial[n, k] StirlingS2[2k, k], {k, 0, n}], {n, 0, 16}]

Maxima

makelist(sum((-1)^(n-k) *binomial(n,k) *stirling2(2*k,k), k,0,n), n,0,12);

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*S(2*k,k).

a(n) ~ c * d^n * (n-1)!, where d = 4/(w*(2-w)) = 6.176554609483480358231680164... and c = exp(w^2/4 - 1) / (Pi * sqrt(2*w * (1-w))) = 0.17569156962762991098958896633434384684339835018075095823375851..., where w = -LambertW(-2*exp(-2))^2 = -A226775. - Vaclav Kotesovec, Mar 30 2018, updated Jul 07 2021