OFFSET
0,4
COMMENTS
Compare the o.g.f. to the curious identity:
1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-(1+n*x)*x).
EXAMPLE
O.g.f: A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 18*x^5 + 96*x^6 + 260*x^7 +...
where
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) +...
simplifies to a power series in x with integer coefficients.
PROG
(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)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 05 2012
STATUS
approved