A187654 Binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

1, 2, 14, 262, 7740, 305536, 15061692, 890220752, 61347750704, 4829414749504, 427559293150976, 42047904926171552, 4547772798257998256, 536504774914535869664, 68557641564333466819744, 9433619169586732241895776

Offset: 0

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A187646.

Programs

Maple

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

Mathematica

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

Maxima

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

PARI

a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(2*k, k, 1))); \\ Michel Marcus, Aug 03 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*s(2k,k).

a(n) ~ exp((2*c-1)/(8*c^2)) * abs(Stirling1(2*n,n)) ~ 2^(3*n-1) * n^n * exp((2*c-1)/(8*c^2)-n) * c^(2*n) / (sqrt(Pi*n*(c-1)) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, May 21 2014