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 A218684 O.g.f.: Sum_{n>=0} (1+n^2*x)^n * x^n/n! * exp(-(1+n^2*x)*x). 3

%I #7 Dec 25 2012 10:11:55

%S 1,0,1,2,7,18,96,260,1851,5270,46515,137942,1447202,4433772,53787706,

%T 169169912,2326986783,7477418982,114916173009,375898894514,

%U 6380455164161,21185872231238,393499602818322,1323362744628080,26691270481453228,90755667374332324

%N O.g.f.: Sum_{n>=0} (1+n^2*x)^n * x^n/n! * exp(-(1+n^2*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 + 2*x^3 + 7*x^4 + 18*x^5 + 96*x^6 + 260*x^7 +...

%e where

%e A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^2*x)^2*x^2/2!*exp(-(1+2^2*x)*x) + (1+3^2*x)^3*x^3/3!*exp(-(1+3^2*x)*x) + (1+4^2*x)^4*x^4/4!*exp(-(1+4^2*x)*x) + (1+5^2*x)^5*x^5/5!*exp(-(1+5^2*x)*x) +...

%e simplifies to a power series in x with integer coefficients.

%o (PARI) {a(n)=polcoeff(sum(k=0,n,(1+k^2*x)^k*x^k/k!*exp(-x*(1+k^2*x)+x*O(x^n))),n)}

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

%Y Cf. A218686, A218685, A218687, A185040.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 05 2012

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