|
%I #5 Nov 04 2012 20:26:36
%S 1,1,3,14,79,516,3802,30668,268815,2522594,25201736,266014607,
%T 2953171684,34326755191,416313253084,5251970372080,68737673434847,
%U 931207966502919,13031639620371226,188051624603419973,2793741995189126920,42668132798523737471,669061042470049870917
%N O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(2*n) * x^n/n! * exp(-n*x*(1+n*x)^2).
%C Compare o.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 + 3*x^2 + 14*x^3 + 79*x^4 + 516*x^5 + 3802*x^6 +...
%e where
%e A(x) = 1 + (1+x)^2*x*exp(-x*(1+x)^2) + 2^2*(1+2*x)^4*x^2/2!*exp(-2*x*(1+2*x)^2) + 3^3*(1+3*x)^6*x^3/3!*exp(-3*x*(1+3*x)^2) + 4^4*(1+4*x)^8*x^4/4!*exp(-4*x*(1+4*x)^2) + 5^5*(1+5*x)^10*x^5/5!*exp(-5*x*(1+5*x)^2) +...
%e simplifies to a power series in x with integer coefficients.
%o (PARI) {a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(2*k)*x^k/k!*exp(-k*x*(1+k*x)^2+x*O(x^n)));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A218670, A218678, A218679.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 04 2012
| 