A187649 Alternated cumulative sums of the central Stirling numbers of the second kind.

1, 0, 7, 83, 1618, 40907, 1282745, 48046535, 2093717518, 104081678237, 5813503286418, 360468997583868, 24569735593174392, 1825998838660375668, 146956989225714933732, 12732911083544911106268, 1181728606386262922675817, 116962289970625115673808638

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..345

Formula

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

a(n) ~ 2^(2*n) * n^(n-1/2) / (sqrt(2*Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, May 30 2025

Maple

seq(sum((-1)^(n-k)*combinat[stirling2](2*k, k), k=0..n), n=0..12);
# Alternative:
L:= [seq((-1)^n*Stirling2(2*n, n), n=0..50)]:
P:= ListTools:-PartialSums(L):
seq((-1)^(n+1)*P[n], n=1..51); # Robert Israel, Aug 25 2017

Mathematica

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

Prog

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

Crossrefs

Cf. A008277 (Stirling2 triangle).

Sequence in context: A231259 A396680 A383231 * A368853 A316377 A053128

Adjacent sequences: A187646 A187647 A187648 * A187650 A187651 A187652

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 12 2011

Status

approved