A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).

1, 1, 4, 16, 82, 502, 3574, 29002, 264166, 2668666, 29612014, 358025986, 4684916902, 65966957722, 994546450174, 15984888286642, 272845934899606, 4929166716321706, 93963635086523374, 1884915966747571906, 39691711412770983622, 875410001054417122042, 20180907494704416823774

Offset: 0

Author

Paul D. Hanna, Jan 09 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 82*x^4 + 502*x^5 + 3574*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(2+x)*(2+2*x)/((1+2*x+x^2)*(1+2*x+2*x^2)) + x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+3*x+2*x^2)*(1+3*x+3*x^2)) + x^4*(4+x)*(4+2*x)*(4+3*x)*(4+4*x)/((1+4*x+x^2)*(1+4*x+2*x^2)*(1+4*x+3*x^2)*(1+4*x+4*x^2)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A204064, A187741, A187742, A220181.

Programs

PARI

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (m+k*x)/(1+m*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ exp(1/2) * n! * n/2. - Vaclav Kotesovec, Nov 02 2014