A218671 O.g.f.: Sum_{n>=0} n^(2*n) * (1+n*x)^n * x^n/n! * exp(-n^2*x*(1+n*x)).

1, 1, 8, 120, 2635, 76503, 2764957, 119634152, 6030195490, 347037131298, 22453144758980, 1613322276606404, 127466755375275614, 10983423290600347408, 1025046637630590359928, 103004615955568528609200, 11088429267977228122393005, 1273093489376335864500416685

Offset: 0

Author

Paul D. Hanna, Nov 04 2012

Keywords

• nonn

Comments

Compare g.f. to the curious identity:

1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).

Examples

			O.g.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 2635*x^4 + 76503*x^5 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^4*(1+2*x)^2*x^2/2!*exp(-2^2*x*(1+2*x)) + 3^6*(1+3*x)^3*x^3/3!*exp(-3^2*x*(1+3*x)) + 4^8*(1+4*x)^4*x^4/4!*exp(-4^2*x*(1+4*x)) + 5^10*(1+5*x)^5*x^5/5!*exp(-5^2*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A007820, A218670, A218672.

Programs

PARI

{a(n)= my(A=sum(k=0, n, k^(2*k)*(1+k*x)^k*x^k/k!*exp(-k^2*x*(1+k*x)+x*O(x^n)))); polcoef(A, n)}
for(n=0,30,print1(a(n),", "))