A209778 - OEIS

%I #16 Nov 02 2014 15:27:23

%S 1,1,1,3,5,19,49,203,733,3315,15241,76731,419973,2375027,14842721,

%T 94159595,655550445,4632480883,35405788601,276183156827,2295741573013,

%U 19588533436019,175928886218769,1628494746863243,15721340742796029,156753433757122035,1619488446357906409

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

%C Compare to the identity:

%C 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).

%H Vaclav Kotesovec, <a href="/A209778/b209778.txt">Table of n, a(n) for n = 0..200</a>

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 19*x^5 + 49*x^6 + 203*x^7 +...

%e where

%e 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)) +...

%o (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)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A208237, A221371.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 19 2013