A243227 G.f.: Sum_{n>=0} n^(2*n) * x^n / (1 + n^2*x)^n.

1, 1, 15, 602, 46620, 5921520, 1118557440, 294293759760, 102896614941120, 46150861752777600, 25832386565857872000, 17651395149921751680000, 14460364581345685626624000, 13990151265412450143375360000, 15782226575197809064309171200000, 20533602558350213132577801792768000

Offset: 0

Author

Paul D. Hanna, Aug 21 2014

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

Cf. A187742, A247238.

Programs

Mathematica

Flatten[{1, Table[(n-1)! * StirlingS2[2*n+1, n],{n,1,20}]}] (* Vaclav Kotesovec, Nov 05 2014 *)

PARI

{a(n)=polcoeff( sum(m=0, n, m^(2*m)*x^m/(1+m^2*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=if(n==0,1,sum(k=0, n-1, (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n)))}
for(n=0,20,print1(a(n),", "))

PARI

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,1, (n-1)! * Stirling2(2*n+1, n) )}
for(n=0, 20, print1(a(n), ", "))

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))}
for(n=0, 20, print1(a(n), ", "))

Formula

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.

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

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

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