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A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%I A128583 #17 Feb 16 2025 08:33:05
%S A128583 1,1,1,2,1,2,1,1,1,0,3,1,1,1,2,2,1,2,0,1,2,1,0,1,2,3,0,1,1,1,3,2,1,1,
%T A128583 1,1,2,0,2,1,2,0,1,0,1,4,1,2,0,1,2,1,2,1,1,3,0,1,2,3,1,0,1,0,0,2,2,1,
%U A128583 1,2,2,1,1,2,0,1,2,0,1,1,6,1,1,1,0,2,1
%N A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%C A128583 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A128583 G. C. Greubel, <a href="/A128583/b128583.txt">Table of n, a(n) for n = 0..1000</a>
%H A128583 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128583 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128583 Expansion of q^(-5/24) * eta(q^2) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
%F A128583 Euler transform of period 12 sequence [ 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 1, -2, ...].
%F A128583 G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 6^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229723.
%F A128583 a(n) = A128582(4*n) = A259895(3*n) = A260118(4*n). 2 * a(n) = A190615(12*n + 2). - _Michael Somos_, Nov 15 2015
%F A128583 -2 * a(n) = A128580(12*n + 2). - _Michael Somos_, Dec 22 2016
%e A128583 G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 3*x^10 + ...
%e A128583 G.f. = q^5 + q^29 + q^53 + 2*q^77 + q^101 + 2*q^125 + q^149 + q^173 + ...
%t A128583 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] QPochhammer[ x^6]^2 / (QPochhammer[ x] QPochhammer[ x^12] ), {x, 0, n}];
%t A128583 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];
%o A128583 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};
%Y A128583 Cf. A128580, A128582, A190615, A229723, A259895, A260118.
%K A128583 nonn
%O A128583 0,4
%A A128583 _Michael Somos_, Mar 11 2007