%I #16 Feb 04 2017 21:53:49
%S 1,8,232,12160,929376,93590784,11709432064,1751777730560,
%T 305065968649728,60623947402670080,13538933075023376384,
%U 3356940619048979988480,915040828127405123420160,271974910674004076827115520,87543520972441760055430348800,30337462571518006406505729884160
%N O.g.f.: Sum_{n>=0} (2*n+1)^(2*n) * exp(-(2*n+1)^2*x) * x^n / n!.
%H G. C. Greubel, <a href="/A222525/b222525.txt">Table of n, a(n) for n = 0..290</a>
%F a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k) * x^k / (1 + (2*k+1)^2*x)^(k+1).
%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n).
%F a(n) = Sum_{k=0..n} 2^(n+k) * binomial(2*n,n+k) * stirling2(n+k,n). - _Vaclav Kotesovec_, May 13 2014
%F a(n) ~ 2^(4*n) * n^(n-1/2) / (sqrt(Pi*r*(1-r)) * exp(n) * (r*(2-r))^n), where r = -LambertW(-2*exp(-2)) = 0.4063757399599... (see A226775 = -r). - _Vaclav Kotesovec_, May 13 2014
%e O.g.f.: A(x) = 1 + 8*x + 232*x^2 + 12160*x^3 + 929376*x^4 + 93590784*x^5 +...
%e where
%e A(x) = exp(-x) + 3^2*exp(-3^2*x)*x + 5^4*exp(-5^2*x)*x^2/2! + 7^6*exp(-7^2*x)*x^3/3! + 9^8*exp(-9^2*x)*x^4/4! + 11^10*exp(-11^2*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(2*k+1)^(2*n),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, May 13 2014 *)
%t Table[Sum[2^k*Binomial[2*n,k]*StirlingS2[k,n],{k,n,2*n}],{n,0,20}] (* _Vaclav Kotesovec_, May 13 2014 *)
%o (PARI) {a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A220955, A217900, A007820, A242373.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 24 2013