A231291 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x) / (1 - x - k*x^2).

1, 1, 3, 9, 29, 99, 355, 1333, 5213, 21163, 88899, 385413, 1720637, 7894827, 37166563, 179254501, 884548253, 4460597131, 22962705027, 120557527941, 644952640253, 3512995320939, 19468234666531, 109694091843109, 628027149163613, 3651429293510731, 21547912967252163

Offset: 0

Author

Paul D. Hanna, Nov 06 2013

Keywords

• nonn

Comments

Compare to the identity:

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 29*x^4 + 99*x^5 + 355*x^6 + 1333*x^7 +...
where
A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(1+2*x)/((1-x-x^2)*(1-x-2*x^2)) + x^3*(1+x)*(1+2*x)*(1+3*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)) + x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)*(1-x-4*x^2)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A204064, A231274.

Programs

PARI

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