A187652 Alternated binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

1, 0, 10, 194, 5932, 237624, 11820780, 702992968, 48662470640, 3843811669088, 341207224961856, 33627579102171680, 3643463136559851440, 430456189350273371648, 55075003474909952394848, 7586546772496980353804704

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Crossrefs

Cf. A187646.

Programs

Maple

seq(sum((-1)^(n-k)*binomial(n,k)*abs(combinat[stirling1](2*k,k)),k=0..n),n=0..12);

Mathematica

Table[Sum[(-1)^(n - k)Binomial[n, k]Abs[StirlingS1[2k, k]], {k, 0, n}], {n, 0, 15}]

Maxima

makelist(sum((-1)^(n-k)*binomial(n,k)*abs(stirling1(2*k,k)),k,0,n),n,0,12);

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*abs(Stirling1(2*k,k)).

a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), where w = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... and c = exp((1-2*w)/(8*w^2)) / (2^(3/2)*Pi*sqrt(w-1)) = exp(-1/d) / (2^(3/2)*Pi*sqrt(w-1)) = 0.11686978539934159049334861225275481804523808136863346883911376048... - Vaclav Kotesovec, Jul 05 2021, updated May 25 2025