%I #5 Nov 05 2012 18:48:02
%S 1,1,2,15,107,1164,13932,207527,3424441,65365273,1366815507,
%T 31899555046,806153628997,22260455705106,659196741236329,
%U 21028295211402871,713819243969142111,25836118882427921161,988875977638287049631,40043648314495526922945
%N O.g.f.: Sum_{n>=0} n^n * (1+n^2*x)^n * x^n/n! * exp(-n*(1+n^2*x)*x).
%C Compare the o.g.f. to the curious identity:
%C 1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-(1+n*x)*x).
%e O.g.f: A(x) = 1 + x + 2*x^2 + 15*x^3 + 107*x^4 + 1164*x^5 + 13932*x^6 +...
%e where
%e 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) +...
%e simplifies to a power series in x with integer coefficients.
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A218684, A218687, A218685.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 05 2012