A187653 - OEIS

%I #20 Oct 19 2024 08:33:23

%S 1,2,10,115,2108,52006,1606229,59550709,2575966264,127343893378,

%T 7081926869746,437585883729512,29740614295527535,2205002457135885616,

%U 177099066222770055407,15317784128757306540986,1419476705128570400447376

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

%H Vincenzo Librandi, <a href="/A187653/b187653.txt">Table of n, a(n) for n = 0..100</a>

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

%F 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

%F 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

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

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

%o (Maxima) makelist(sum(binomial(n,k)*stirling2(2*k,k),k,0,n),n,0,12);

%o (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)

%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 02 2013

%Y Cf. A007820.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Mar 12 2011