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URL: https://oeis.org/A206141

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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.
2
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).
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved