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%I #5 Nov 05 2012 18:47:36
%S 1,0,1,6,34,270,3415,31230,681026,6949920,230637870,2546120514,
%T 119281951006,1394371349490,87612425583018,1069010047029672,
%U 86763885548985810,1094149501538197236,111443560982774811439,1442387644419293694144,180179254059921915232864
%N O.g.f.: Sum_{n>=0} (1+n^3*x)^n * x^n/n! * exp(-(1+n^3*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 + 6*x^3 + 34*x^4 + 270*x^5 + 3415*x^6 +...
%e where
%e A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^3*x)^2*x^2/2!*exp(-(1+2^3*x)*x) + (1+3^3*x)^3*x^3/3!*exp(-(1+3^3*x)*x) + (1+4^3*x)^4*x^4/4!*exp(-(1+4^3*x)*x) + (1+5^3*x)^5*x^5/5!*exp(-(1+5^3*x)*x) +...
%e simplifies to a power series in x with integer coefficients.
%o (PARI) {a(n)=polcoeff(sum(k=0,n,(1+k^3*x)^k*x^k/k!*exp(-x*(1+k^3*x)+x*O(x^n))),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A218687, A218684, A218686.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 05 2012
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