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.
Compare also 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..268
FORMULA
a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Oct 31 2014
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 104*x^4 + 736*x^5 + 6232*x^6 +...
where
A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(2+x)/((1-x-x^2)*(1-2*x-x^2)) + x^3*(1+x)*(2+x)*(3+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)) + x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1-x-x^2)*(1-2*x-x^2)*(1-3*x-x^2)*(1-4*x-x^2)) +...
PROG
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k+x)/(1-k*x-x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 06 2013
STATUS
approved