%I #8 Aug 30 2024 14:54:59
%S 1,1,8,120,2635,76503,2764957,119634152,6030195490,347037131298,
%T 22453144758980,1613322276606404,127466755375275614,
%U 10983423290600347408,1025046637630590359928,103004615955568528609200,11088429267977228122393005,1273093489376335864500416685
%N O.g.f.: Sum_{n>=0} n^(2*n) * (1+n*x)^n * x^n/n! * exp(-n^2*x*(1+n*x)).
%C Compare g.f. to the curious identity:
%C 1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).
%e O.g.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 2635*x^4 + 76503*x^5 +...
%e where
%e 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)) +...
%e simplifies to a power series in x with integer coefficients.
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A007820, A218670, A218672.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 04 2012