%I #11 Dec 04 2017 03:01:10

%S 1,-3,3,-6,9,0,24,-78,63,-102,0,0,432,-1398,2262,0,2961,0,7752,-25086,

%T 0,-65676,0,0,139104,-225075,728358,-589254,1906866,0,0,-8077614,

%U 6534927,0,0,0,44791056,-144946902,234529014,-379475916,0,0,1607485776,-2600966622,0

%N a(n) = Fibonacci(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

%C Compare g.f. to the Lambert series of A132973:

%C 1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).

%H G. C. Greubel, <a href="/A205970/b205970.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 - 3*Sum_{n>=0} Fibonacci(6*n+1)*x^(6*n+1)/(1 + Lucas(6*n+1) * x^(6*n+1) - x^(12*n+2)) - Fibonacci(6*n+5)*x^(6*n +5)/(1 + Lucas(6*n+5) * x^(6*n+5) - x^(12*n+10)).

%e G.f.: A(x) = 1 - 3*x + 3*x^2 - 6*x^3 + 9*x^4 + 24*x^6 - 78*x^7 + 63*x^8 +...

%e where A(x) = 1 - 1*3*x + 1*3*x^2 - 2*3*x^3 + 3*3*x^4 + 8*3*x^6 - 13*6*x^7 + 21*3*x^8 +...+ Fibonacci(n)*A132973(n)*x^n +...

%e The g.f. is also given by the identity:

%e A(x) = 1 - 3*( 1*x/(1+x-x^2) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 89*x^11/(1+199*x^11-x^22) + 233*x^13/(1+521*x^13-x^26) - 1597*x^17/(1+3571*x^17-x^34) +...).

%t A132973:= CoefficientList[Series[(-1)^(-1/4)*EllipticTheta[2, 0, I*Sqrt[q]]^3/EllipticTheta[2, 0, I*Sqrt[q^3]]/4, {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n]*A132973[[n + 1]]], {n, 0, 50}] (* _G. C. Greubel_, Dec 03 2017 *)

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(1 - 3*sum(m=0,n, fibonacci(6*m+1)*x^(6*m+1)/(1+Lucas(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - fibonacci(6*m+5)*x^(6*m+5)/(1+Lucas(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ),n)}

%o for(n=0,61,print1(a(n),", "))

%Y Cf. A132973, A205969, A205971, A203847.

%Y Cf. A209450 (Pell variant).

%K sign

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012