A205969 a(n) = Fibonacci(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

1, -2, 4, -4, 6, 0, 32, -52, 84, -68, 0, 0, 288, -932, 3016, 0, 1974, 0, 10336, -16724, 0, -43784, 0, 0, 185472, -150050, 971144, -392836, 1271244, 0, 0, -5385076, 8713236, 0, 0, 0, 29860704, -96631268, 312705352, -252983944, 0, 0, 2143314368, -1733977748, 0

Offset

0,2

Comments

Compare g.f. to the Lambert series of A113973: 1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

G.f.: 1 - 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

Example

G.f.: A(x) = 1 - 2*x + 4*x^2 - 4*x^3 + 6*x^4 + 32*x^6 - 52*x^7 + 84*x^8 +...
where A(x) = 1 - 1*2*x + 1*4*x^2 - 2*2*x^3 + 3*2*x^4 + 8*4*x^6 - 13*4*x^7 + 21*4*x^8 +...+ Fibonacci(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 1*x/(1+x-x^2) - 1*x^2/(1-3*x^2+x^4) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 21*x^8/(1-47*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Mathematica

A113973:= CoefficientList[Series[EllipticTheta[3, q^3]^3/EllipticTheta[3, 0, q], {q, 0, 75}], q]; Table[If[n == 1, 1, Fibonacci[n-1]*A113973[[n]] ], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 2*sum(m=1, n, fibonacci(m)*kronecker(m, 3)*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A113973, A205966, A205968, A205970, A203847, A000204 (Lucas).

Cf. A209449 (Pell variant).

Sequence in context: A223227 A160904 A306340 * A326771 A049782 A372785

Adjacent sequences: A205966 A205967 A205968 * A205970 A205971 A205972

Keyword

sign

Author

Paul D. Hanna, Feb 04 2012

Status

approved