A218668 - OEIS

%I #7 Oct 30 2014 17:22:34

%S 1,0,1,3,22,161,1546,18857,270320,4471693,85455574,1865128265,

%T 45735737037,1247518965519,37654095184226,1250673144714138,

%U 45415758777730668,1792734161930717221,76595370803745016626,3529261203030717032927,174742139545017029583279

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

%H Vaclav Kotesovec, <a href="/A218668/b218668.txt">Table of n, a(n) for n = 0..290</a>

%e O.g.f.: A(x) = 1 + x^2 + 3*x^3 + 22*x^4 + 161*x^5 + 1546*x^6 + 18857*x^7 +...

%e where

%e A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-4*x)^2/2!*exp(-x/(1-4*x)) + x^3/(1-9*x)^3/3!*exp(-x/(1-9*x)) + x^4/(1-16*x)^4/4!*exp(-x/(1-16*x)) + x^5/(1-25*x)^5/5!*exp(-x/(1-25*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^2*X)^k*x^k/k!*exp(-X/(1-k^2*X)));polcoeff(A,n)}

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

%Y Cf. A218667, A218669, A218670, A217900.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 04 2012