A187653 Binomial cumulative sums of the central Stirling numbers of the second kind (A007820).

1, 2, 10, 115, 2108, 52006, 1606229, 59550709, 2575966264, 127343893378, 7081926869746, 437585883729512, 29740614295527535, 2205002457135885616, 177099066222770055407, 15317784128757306540986, 1419476705128570400447376

Offset

0,2

Links

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

Formula

a(n) = sum(binomial(n,k)S(2k,k)),k=0..n)

a(n) ~ exp(c*(2-c)/4) * StirlingS2(2*n,n) ~ 2^(2*n-1/2)*n^(n-1/2)/(sqrt(Pi*(1-c))*exp(n-c*(2-c)/4)*(c*(2-c))^n), where c = - LambertW(-2/exp(2)) = 0.406375739959959907676958... - Vaclav Kotesovec, Jan 02 2013

O.g.f.: Sum_{n>=0} n^(2*n)/n! * x^n/(1-x)^(n+1) * exp(-n^2*x/(1-x)). - Paul D. Hanna, Jan 02 2013

Maple

seq(sum(binomial(n, k)*combinat[stirling2](2*k, k), k=0..n), n=0..12);

Mathematica

Table[Sum[Binomial[n, k]StirlingS2[2k, k], {k, 0, n}], {n, 0, 16}]

Prog

(Maxima) makelist(sum(binomial(n, k)*stirling2(2*k, k), k, 0, n), n, 0, 12);
(PARI) a(n)=polcoeff(sum(m=0, n, m^(2*m)/m!*x^m/(1-x)^(m+1)*exp(-m^2*x/(1-x+x*O(x^n)))), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 02 2013

Crossrefs

Cf. A007820.

Sequence in context: A113089 A054928 A132522 * A131811 A261496 A347014

Adjacent sequences: A187650 A187651 A187652 * A187654 A187655 A187656

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 12 2011

Status

approved