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A128591 Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

%I A128591 #13 Feb 16 2025 08:33:05
%S A128591 1,2,1,1,1,1,2,1,2,1,1,3,0,0,1,2,2,1,1,1,1,2,3,1,1,0,2,1,1,2,0,2,0,2,
%T A128591 1,0,4,2,0,1,1,2,1,2,2,1,2,0,1,1,2,0,1,1,1,2,2,2,0,1,1,3,1,1,0,1,4,1,
%U A128591 2,1,0,4,0,0,1,1,2,1,2,1,1,0,1,1,1,2,3
%N A128591 Expansion of  f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%C A128591 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A128591 G. C. Greubel, <a href="/A128591/b128591.txt">Table of n, a(n) for n = 0..1000</a>
%H A128591 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128591 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128591 Expansion of chi(x) * psi(x) * psi(-x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Nov 15 2015
%F A128591 Expansion of  q^(-11/24) * eta(q^2)^4 * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
%F A128591 Euler transform of period 12 sequence [ 2, -2, 1, -1, 2, -2, 2, -1, 1, -2, 2, -2, ...].
%F A128591 a(n) = A128582(4*n + 1).
%F A128591 2 * a(n) = A257920(3*n + 1). - a(n) = A260118(4*n + 1). 2 * a(n) = A257921(6*n + 2). -2 * a(n) = A128580(12*n + 5) = A190615(12*n + 5). - _Michael Somos_, Nov 15 2015
%e A128591 G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + x^10 + 3*x^11 + ...
%e A128591 G.f. = q^11 + 2*q^35 + q^59 + q^83 + q^107 + q^131 + 2*q^155 + q^179 + 2*q^203 + ...
%t A128591 a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* _Michael Somos_, Nov 15 2015 *)
%o A128591 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff(  eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^12 + A)  / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};
%Y A128591 Cf. A128580, A128582, A190615, A257920, A257921,A260118.
%K A128591 nonn
%O A128591 0,2
%A A128591 _Michael Somos_, Mar 11 2007