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

0,3

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).

Links

Table of n, a(n) for n=0..29.

Example

 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, ...].

Prog

 (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), ", "))

Crossrefs

Cf. A002203 (Co.Pell), A206141.

Sequence in context: A130648 A061812 A009452 * A281972 A355869 A343553

Adjacent sequences: A206138 A206139 A206140 * A206142 A206143 A206144

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2012

Status

approved