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%I #14 Jul 18 2018 06:43:37
%S 1,-2,4,-4,6,0,32,-52,84,-68,0,0,288,-932,3016,0,1974,0,10336,-16724,
%T 0,-43784,0,0,185472,-150050,971144,-392836,1271244,0,0,-5385076,
%U 8713236,0,0,0,29860704,-96631268,312705352,-252983944,0,0,2143314368,-1733977748,0
%N 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.
%C Compare g.f. to the Lambert series of A113973: 1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).
%H G. C. Greubel, <a href="/A205969/b205969.txt">Table of n, a(n) for n = 0..2500</a>
%F G.f.: 1 - 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e 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 +...
%e 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 +...
%e The g.f. is also given by the identity:
%e 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) +...).
%e The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%t 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 *)
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {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)}
%o for(n=0,60,print1(a(n),", "))
%Y Cf. A113973, A205966, A205968, A205970, A203847, A000204 (Lucas).
%Y Cf. A209449 (Pell variant).
%K sign
%O 0,2
%A _Paul D. Hanna_, Feb 04 2012
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