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%I #36 Mar 12 2021 22:24:48
%S 1,3,4,4,4,7,8,4,5,8,12,8,4,12,8,8,9,12,16,4,12,15,8,8,8,20,20,8,8,12,
%T 16,16,8,15,20,12,12,16,16,12,13,24,20,8,8,24,24,8,16,12,28,16,12,28,
%U 8,20,13,20,28,16,20,24,16,8,8,27,36,12,16,28,24,12
%N Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%H G. C. Greubel, <a href="/A259884/b259884.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(x) * c(x) * (1/3) * x^(-1/3) in powers of x where phi() is a Ramanujan theta function (A000122) and c() is a cubic AGM theta function (A005882).
%F Expansion of q^(-1/3) * eta(q^2)^5 * eta(q^3)^3 / (eta(q)^3 * eta(q^4)^2) in powers of q.
%F Euler transform of period 12 sequence [3, -2, 0, 0, 3, -5, 3, 0, 0, -2, 3, -3, ...].
%e G.f. = 1 + 3*x + 4*x^2 + 4*x^3 + 4*x^4 + 7*x^5 + 8*x^6 + 4*x^7 + 5*x^8 + ...
%e G.f. = q + 3*q^4 + 4*q^7 + 4*q^10 + 4*q^13 + 7*q^16 + 8*q^19 + 4*q^22 + ...
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[q^(-1/3)* eta[q^2]^5*eta[q^3]^3/(eta[q]^3*eta[q^4]^2), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 16 2018 *)
%t a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]*QPochhammer[q^3]^3 /QPochhammer[q], {q,0,n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 18 2018 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^3 / (eta(x + A)^3 * eta(x^4 + A)^2), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q^2)^5*eta(q^3)^3/(eta(q)^3*eta(q^4)^2)) \\ _Altug Alkan_, Mar 18 2018
%Y Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%Y Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%K nonn
%O 0,2
%A _Michael Somos_, Jul 07 2015
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