A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + n*k*x) / (1 + x + n*k*x^2).

1, 1, 1, 4, 10, 50, 208, 1290, 7456, 55982, 411796, 3650514, 32484460, 332970374, 3468625588, 40420787250, 481757564956, 6295577910182, 84407459209876, 1223095585594674, 18208380720893980, 289843786627539014, 4741844351895315028, 82269590167564595250

Offset: 0

Author

Paul D. Hanna, Jan 10 2013

Keywords

• nonn

Comments

Compare to the identity:

Sum_{n>=0} x^n * Product_{k=1..n} (1 + t*k*x) / (1 + x + t*k*x^2) = (1+x)/(1-t*x^2).

Compare to the g.f. of A187741:

Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n = 1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.

Limit n->infinity (a(n)/n!)^(1/n) = 1/(2*log(2)). - Vaclav Kotesovec, Nov 03 2014

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 50*x^5 + 208*x^6 + 1290*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*1*x)*(1+2*2*x)/((1+x+2*1*x^2)*(1+x+2*2*x^2)) + x^3*(1+3*1*x)*(1+3*2*x)*(1+3*3*x)/((1+x+3*1*x^2)*(1+x+3*2*x^2)*(1+x+3*3*x^2)) + x^4*(1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)/((1+x+4*1*x^2)*(1+x+4*2*x^2)*(1+x+4*3*x^2)*(1+x+4*4*x^2)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A204064, A187741, A204066.

Programs

PARI

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