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

Author

Emanuele Munarini, Mar 12 2011

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A007820.

Programs

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}]

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

Formula

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

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