OFFSET
0,3
COMMENTS
Compare o.g.f. to the curious identity:
1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).
EXAMPLE
O.g.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 79*x^4 + 516*x^5 + 3802*x^6 +...
where
A(x) = 1 + (1+x)^2*x*exp(-x*(1+x)^2) + 2^2*(1+2*x)^4*x^2/2!*exp(-2*x*(1+2*x)^2) + 3^3*(1+3*x)^6*x^3/3!*exp(-3*x*(1+3*x)^2) + 4^4*(1+4*x)^8*x^4/4!*exp(-4*x*(1+4*x)^2) + 5^5*(1+5*x)^10*x^5/5!*exp(-5*x*(1+5*x)^2) +...
simplifies to a power series in x with integer coefficients.
PROG
(PARI) {a(n)=local(A=1+x); A=sum(k=0, n, k^k*(1+k*x)^(2*k)*x^k/k!*exp(-k*x*(1+k*x)^2+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 04 2012
STATUS
approved