A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

1, -9, 54, -45, -1404, 6264, 1890, -76050, 187272, -8865, -1540944, 6200280, -1621620, -51195330, 109055700, 42125400, -868685040, 2946297888, 74093670, -21584605122, 44912353824, -17376284250, -302040439920, 1069478852112, 249392931480, -7095191496489

Offset

0,2

Comments

Compare the g.f. to the Lambert series of A109041:

1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

Example

G.f.: A(x) = 1 - 9*x + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Mathematica

A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)

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(1 - 9*sum(m=1, n, Pell(m)*kronecker(m, 3)*m^2*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.

Sequence in context: A336184 A326606 A052108 * A259316 A224484 A225791

Adjacent sequences: A209450 A209451 A209452 * A209454 A209455 A209456

Keyword

sign

Author

Paul D. Hanna, Mar 10 2012

Status

approved