%I #20 Nov 29 2014 12:56:46

%S 1,1,15,602,46620,5921520,1118557440,294293759760,102896614941120,

%T 46150861752777600,25832386565857872000,17651395149921751680000,

%U 14460364581345685626624000,13990151265412450143375360000,15782226575197809064309171200000,20533602558350213132577801792768000

%N G.f.: Sum_{n>=0} n^(2*n) * x^n / (1 + n^2*x)^n.

%C Compare to: Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.

%F a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n) for n>0 with a(0)=1.

%F a(n) = (n-1)! * Stirling2(2*n+1, n) for n>0 with a(0)=1.

%F a(n) = (2*n+1)!/n * [x^(2*n+1)] (exp(x) - 1)^n for n>0 with a(0)=1.

%F a(n) ~ 2^(2*n+1) * n^(2*n) / (sqrt(1-c) * exp(2*n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - _Vaclav Kotesovec_, Nov 05 2014

%e G.f.: A(x) = 1 + x + 15*x^2 + 602*x^3 + 46620*x^4 + 5921520*x^5 +...

%e where

%e A(x) = 1 + x/(1+x) + 4^2*x^2/(1+4*x)^2 + 9^3*x^3/(1+9*x)^3 + 16^4*x^4/(1+16*x)^4 + 25^5*x^5/(1+25*x)^5 + 36^6*x^6/(1+36*x)^6 + 49^7*x^7/(1+49*x)^7 +...

%t Flatten[{1, Table[(n-1)! * StirlingS2[2*n+1, n],{n,1,20}]}] (* _Vaclav Kotesovec_, Nov 05 2014 *)

%o (PARI) {a(n)=polcoeff( sum(m=0, n, m^(2*m)*x^m/(1+m^2*x +x*O(x^n))^m), n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n)=if(n==0,1,sum(k=0, n-1, (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n)))}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

%o {a(n) = if(n==0,1, (n-1)! * Stirling2(2*n+1, n) )}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = if(n==0,1,(2*n+1)!/n * polcoeff(((exp(x + x*O(x^(2*n+1))) - 1)^n), 2*n+1))}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A187742, A247238.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 21 2014