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A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

%I A260110 #10 Feb 16 2025 08:33:26
%S A260110 1,-2,0,0,3,-2,0,0,3,-4,0,0,2,-2,0,0,2,-2,0,0,3,-2,0,0,4,-2,0,0,1,-6,
%T A260110 0,0,2,-2,0,0,4,-2,0,0,2,0,0,0,4,-2,0,0,1,-4,0,0,2,-4,0,0,2,-4,0,0,1,
%U A260110 -2,0,0,8,0,0,0,2,-4,0,0,2,-2,0,0,2,-2,0,0,0
%N A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.
%C A260110 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260110 G. C. Greubel, <a href="/A260110/b260110.txt">Table of n, a(n) for n = 0..1000</a>
%H A260110 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260110 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260110 Expansion of q^(-1/6) * eta(q)^2 * eta(q^8) * eta(q^12)^2 / (eta(q^2) * eta(q^4) * eta(q^24)) in powers of q.
%F A260110 Euler transform of period 24 sequence [ -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -2, ...].
%F A260110 a(n) = A134177(3*n) = A190615(3*n) = A229723(6*n + 1). a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A113780(n). a(4*n + 1) = -2 * A260089(n).
%e A260110 G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 3*x^8 - 4*x^9 + 2*x^12 - 2*x^13 + 2*x^16 + ...
%e A260110 G.f. = q - 2*q^7 + 3*q^25 - 2*q^31 + 3*q^49 - 4*q^55 + 2*q^73 - 2*q^79 + ...
%t A260110 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];
%o A260110 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^24 + A)), n))};
%Y A260110 Cf. A134177, A190615, A229723, A260089.
%K A260110 sign
%O A260110 0,2
%A A260110 _Michael Somos_, Jul 16 2015