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%I #4 Nov 04 2012 15:05:55
%S 1,0,1,7,97,1561,41136,1551814,72440460,4281320257,324623105584,
%T 30086950057627,3299720918091511,428431079916572044,
%U 65637957066642609845,11659659637028895337265,2367270866164121777222596,546795407830461739380895161,143176487805296033192642234802
%N O.g.f.: Sum_{n>=0} 1/(1-n^3*x)^n * x^n/n! * exp(-x/(1-n^3*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^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
%e where
%e A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-8*x)^2/2!*exp(-x/(1-8*x)) + x^3/(1-27*x)^3/3!*exp(-x/(1-27*x)) + x^4/(1-64*x)^4/4!*exp(-x/(1-64*x)) + x^5/(1-125*x)^5/5!*exp(-x/(1-125*x)) +...
%e simplifies to a power series in x with integer coefficients.
%o (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k^3*X)^k*x^k/k!*exp(-X/(1-k^3*X)));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A218667, A218668, A218670, A217900.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 04 2012
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