A209778 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k^2*x) / (1 + x + k^2*x^2).

1, 1, 1, 3, 5, 19, 49, 203, 733, 3315, 15241, 76731, 419973, 2375027, 14842721, 94159595, 655550445, 4632480883, 35405788601, 276183156827, 2295741573013, 19588533436019, 175928886218769, 1628494746863243, 15721340742796029, 156753433757122035, 1619488446357906409

Offset: 0

Author

Paul D. Hanna, Jan 19 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).

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 19*x^5 + 49*x^6 + 203*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+x)*(1+4*x)/((1+x+x^2)*(1+x+4*x^2)) + x^3*(1+x)*(1+4*x)*(1+9*x)/((1+x+x^2)*(1+x+4*x^2)*(1+x+9*x^2)) + x^4*(1+x)*(1+4*x)*(1+9*x)*(1+16*x)/((1+x+x^2)*(1+x+4*x^2)*(1+x+9*x^2)*(1+x+16*x^2)) +...
		

Links

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

Crossrefs

Cf. A208237, A221371.

Programs

PARI

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