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A205968 a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.

%I A205968 #14 Jul 16 2018 21:48:57
%S A205968 1,24,216,1776,5256,15120,63936,107328,294840,823344,1496880,2845152,
%T A205968 9334656,12291216,28012608,68251680,110883528,188343792,563167296,
%U A205968 688359840,1493387280,3343696512,5095667232,8368761024,24087248640,28361250600,57633511824,128471514096
%N A205968 a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.
%C A205968 Compare g.f. to the Lambert series of A008655:
%C A205968 1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).
%H A205968 G. C. Greubel, <a href="/A205968/b205968.txt">Table of n, a(n) for n = 0..1000</a>
%F A205968 G.f.: 1 + Sum_{n>=1} 24*Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) + 8*Fibonacci(3*n)*(3*n)^3*x^(3*n)/(1 - Lucas(3*n)*x^(3*n) + (-1)^n*x^(6*n)).
%e A205968 G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
%e A205968 where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...
%t A205968 A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* _G. C. Greubel_, Jul 16 2018 *)
%o A205968 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205968 {a(n)=polcoeff(1 + sum(m=1,n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
%o A205968 for(n=0,40,print1(a(n),", "))
%Y A205968 Cf. A008655, A205967, A205969, A203847, A000204 (Lucas).
%Y A205968 Cf. A209448 (Pell variant).
%K A205968 nonn
%O A205968 0,2
%A A205968 _Paul D. Hanna_, Feb 04 2012