OFFSET
0,3
COMMENTS
Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 + k*x + x^2) = (1+x^2)/(1-x).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..220
FORMULA
a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 174*x^4 + 2262*x^5 + 40894*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+x)*(4+x)/((1+x+x^2)*(1+4*x+x^2)) + x^3*(1+x)*(4+x)*(9+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)*(1+16*x+x^2)) +...
PROG
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k^2+x)/(1+k^2*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 19 2013
STATUS
approved