A218686 O.g.f.: Sum_{n>=0} n^n * (1+n^2*x)^n * x^n/n! * exp(-n*(1+n^2*x)*x).

1, 1, 2, 15, 107, 1164, 13932, 207527, 3424441, 65365273, 1366815507, 31899555046, 806153628997, 22260455705106, 659196741236329, 21028295211402871, 713819243969142111, 25836118882427921161, 988875977638287049631, 40043648314495526922945

Offset: 0

Author

Paul D. Hanna, Nov 05 2012

Keywords

• nonn

Comments

Compare the o.g.f. to the curious identity:

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

Examples

			O.g.f: A(x) = 1 + x + 2*x^2 + 15*x^3 + 107*x^4 + 1164*x^5 + 13932*x^6 +...
where
A(x) = 1 + (1+x)*x*exp(-(1+x)*x) + 2^2*(1+2^2*x)^2*x^2/2!*exp(-2*(1+2^2*x)*x) + 3^3*(1+3^2*x)^3*x^3/3!*exp(-3*(1+3^2*x)*x) + 4^4*(1+4^2*x)^4*x^4/4!*exp(-4*(1+4^2*x)*x) + 5^5*(1+5^2*x)^5*x^5/5!*exp(-5*(1+5^2*x)*x) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218684, A218687, A218685.

Programs

PARI

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