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A205969
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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.
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6
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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
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OFFSET
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0,2
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COMMENTS
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Compare g.f. to the Lambert series of A113973: 1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).
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LINKS
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FORMULA
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G.f.: 1 - 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
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EXAMPLE
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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, ...].
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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