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A233037 Expansion of 3 * q^(1/3) * phi(q) * psi(q^6) / c(q) in powers of x where phi(), psi() are Ramanujan theta functions and c() is a cubic AGM theta function.

%I A233037 #14 Feb 16 2025 08:33:20
%S A233037 1,1,-3,1,5,-10,4,17,-31,9,46,-79,21,112,-183,46,249,-396,98,521,-815,
%T A233037 193,1041,-1599,373,1998,-3031,696,3708,-5567,1262,6694,-9955,2233,
%U A233037 11788,-17393,3872,20313,-29771,6572,34342,-50016,10973,57065,-82654,18030
%N A233037 Expansion of 3 * q^(1/3) * phi(q) * psi(q^6) / c(q) in powers of x where phi(), psi() are Ramanujan theta functions and c() is a cubic AGM theta function.
%C A233037 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A233037 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A233037 G. C. Greubel, <a href="/A233037/b233037.txt">Table of n, a(n) for n = 0..1000</a>
%H A233037 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A233037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A233037 Expansion of q^(-5/12) * eta(q^2)^5 * eta(q^12)^2 / (eta(q) * eta(q^4)^2 * eta(q^3)^3 * eta(q^6)) in powers of q.
%F A233037 Euler transform of period 12 sequence [ 1, -4, 4, -2, 1, 0, 1, -2, 4, -4, 1, 0, ...].
%F A233037 a(n) = A182032(12*n + 5).
%e A233037 G.f. = 1 + x - 3*x^2 + x^3 + 5*x^4 - 10*x^5 + 4*x^6 + 17*x^7 - 31*x^8 + ...
%e A233037 G.f. = q^5 + q^17 - 3*q^29 + q^41 + 5*q^53 - 10*q^65 + 4*q^77 + 17*q^89 + ...
%t A233037 eta[x_] := x^(1/24)*QPochhammer[x]; A233037[n_] := SeriesCoefficient[q^(-5/12)*eta[q^2]^5* eta[q^12]^2/(eta[q]*eta[q^4]^2*eta[q^3]^3*eta[q^6]), {q,0,n}];
%t A233037 Table[A233037[n], {n,0,50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o A233037 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^3 + A)^3 * eta(x^6 + A)), n))}
%Y A233037 Cf. A182032.
%K A233037 sign
%O A233037 0,3
%A A233037 _Michael Somos_, Dec 03 2013