A206141 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - A002203(k)*x^k + (-1)^k*x^(2*k)), where A002203 is the companion Pell numbers.

1, 1, 3, 8, 26, 67, 216, 555, 1704, 4538, 13320, 35376, 103863, 273792, 783694, 2101835, 5905044, 15745360, 44132278, 117267422, 325136638, 868034994, 2379074541, 6337238658, 17347580484, 46039358056, 125056019725, 332678989816, 898361151760, 2382959919616

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).

As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - 2*u^k*x - (u^k*x)^2).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 67*x^5 + 216*x^6 + 555*x^7 +...
where
A(x) = 1 + x/(1-2*x-x^2) + x^2/((1-2*x-x^2)*(1-6*x^2+x^4)) + x^3/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)) + x^4/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)) + x^5/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)*(1-82*x^5-x^10)) +...
The companion Pell numbers begin:
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].
		

Crossrefs

Cf. A002203 (Co.Pell), A206141.

Programs

PARI

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-A002203(k)*x^k+(-1)^k*x^(2*k)+x*O(x^n))),n)}
for(n=0,51,print1(a(n),", "))