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O.g.f.: Sum_{n>=0} n^n * (1+n^3*x)^n * x^n/n! * exp(-n*(1+n^3*x)*x).
3

%I #7 Nov 05 2012 18:48:33

%S 1,1,2,31,398,10476,296407,12613297,592445192,36797742660,

%T 2524966492661,212912151736648,19819138754732997,2155966497948737905,

%U 259256365067737582615,35050667748654756208069,5257919606219599751747894,858816581875175776426876930

%N O.g.f.: Sum_{n>=0} n^n * (1+n^3*x)^n * x^n/n! * exp(-n*(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*x^2 + 31*x^3 + 398*x^4 + 10476*x^5 + 296407*x^6 +...

%e where

%e A(x) = 1 + (1+x)*x*exp(-(1+x)*x) + 2^2*(1+2^3*x)^2*x^2/2!*exp(-2*(1+2^3*x)*x) + 3^3*(1+3^3*x)^3*x^3/3!*exp(-3*(1+3^3*x)*x) + 4^4*(1+4^3*x)^4*x^4/4!*exp(-4*(1+4^3*x)*x) + 5^5*(1+5^3*x)^5*x^5/5!*exp(-5*(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,k^k*(1+k^3*x)^k*x^k/k!*exp(-k*x*(1+k^3*x)+x*O(x^n))),n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A218685, A218686, A218684.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 05 2012