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A205970
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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.
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5
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1, -3, 3, -6, 9, 0, 24, -78, 63, -102, 0, 0, 432, -1398, 2262, 0, 2961, 0, 7752, -25086, 0, -65676, 0, 0, 139104, -225075, 728358, -589254, 1906866, 0, 0, -8077614, 6534927, 0, 0, 0, 44791056, -144946902, 234529014, -379475916, 0, 0, 1607485776, -2600966622, 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 A132973:
1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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 +...
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 +...
The g.f. is also given by the identity:
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) +...).
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MATHEMATICA
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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 *)
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PROG
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(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{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)}
for(n=0, 61, 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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