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A213592 Expansion of q^(-1/3) * phi(q^2) * c(q) / 3 in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

%I A213592 #10 Feb 16 2025 08:33:17
%S A213592 1,1,4,2,6,1,6,2,7,4,8,4,10,2,10,0,9,6,12,6,10,1,14,4,16,6,8,8,12,2,
%T A213592 12,0,20,7,20,6,10,4,20,6,11,8,16,8,20,4,14,0,20,10,12,8,26,2,22,6,15,
%U A213592 10,20,12,18,0,28,0,20,9,20,14,16,6,10,6,24,12,32
%N A213592 Expansion of q^(-1/3) * phi(q^2) * c(q) / 3 in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%C A213592 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213592 G. C. Greubel, <a href="/A213592/b213592.txt">Table of n, a(n) for n = 0..1000</a>
%H A213592 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213592 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213592 Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
%F A213592 Euler transform of period 24 sequence [ 1, 3, -2, -2, 1, 0, 1, 0, -2, 3, 1, -5, 1, 3, -2, 0, 1, 0, 1, -2, -2, 3, 1, -3, ...].
%F A213592 a(16*n + 15) = 0. a(4*n + 1) = a(n).
%e A213592 1 + x + 4*x^2 + 2*x^3 + 6*x^4 + x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 4*x^9 + ...
%e A213592 q + q^4 + 4*q^7 + 2*q^10 + 6*q^13 + q^16 + 6*q^19 + 2*q^22 + 7*q^25 + ...
%t A213592  QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^4]^5)/( QP[q]*QP[q^2]^2*QP[q^8]^2), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 07 2018 *)
%o A213592 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}
%K A213592 nonn
%O A213592 0,3
%A A213592 _Michael Somos_, Jun 15 2012