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A205972 a(n) = Fibonacci(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

%I A205972 #11 Dec 04 2017 03:01:21
%S A205972 1,-6,12,-12,-18,0,96,-156,252,-204,0,0,-864,-2796,9048,0,-5922,0,
%T A205972 31008,-50172,0,-131352,0,0,556416,-450150,2913432,-1178508,-3813732,
%U A205972 0,0,-16155228,26139708,0,0,0,-89582112,-289893804,938116056,-758951832,0,0,6429943104
%N A205972 a(n) = Fibonacci(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.
%C A205972 Compare the g.f. to the Lambert series of A122859:
%C A205972 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).
%H A205972 G. C. Greubel, <a href="/A205972/b205972.txt">Table of n, a(n) for n = 0..1000</a>
%F A205972 G.f.: 1 - 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205972 G.f.: A(x) = 1 - 6*x + 12*x^2 - 12*x^3 - 18*x^4 + 96*x^6 - 156*x^7 +...
%e A205972 where A(x) = 1 - 1*6*x + 1*12*x^2 - 2*6*x^3 - 3*6*x^4 + 8*12*x^6 - 13*12*x^7 + 21*12*x^8 - 34*6*x^9 +...+ Fibonacci(n)*A122859(n)*x^n +...
%e A205972 The g.f. is also given by the identity:
%e A205972 A(x) = 1 - 6*( 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) +...).
%e A205972 The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%t A205972 A122859:= CoefficientList[Series[EllipticTheta[3, 0, -q]^3/EllipticTheta[3, 0, -q^3], {q, 0, 60}], q]; Table[If[n == 1, 1, Fibonacci[n - 1]*A122859[[n]]], {n, 1, 50}] (* _G. C. Greubel_, Dec 03 2017 *)
%o A205972 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205972 {a(n)=polcoeff(1 - 6*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)}
%o A205972 for(n=0,40,print1(a(n),", "))
%Y A205972 Cf. A122859, A205971, A205973, A205966, A205969, A203847, A000204 (Lucas).
%Y A205972 Cf. A209452 (Pell variant).
%K A205972 sign
%O A205972 0,2
%A A205972 _Paul D. Hanna_, Feb 04 2012