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A260221 Expansion of phi(x^3)^2 / f(x) in powers of x where phi(), f() are Ramanujan theta functions.

%I A260221 #12 Feb 16 2025 08:33:26
%S A260221 1,-1,2,1,1,1,3,1,2,2,2,4,5,3,7,8,7,7,9,10,11,12,14,17,19,18,24,26,26,
%T A260221 31,36,38,41,45,50,57,61,63,75,83,86,93,106,115,123,134,146,162,173,
%U A260221 183,206,225,237,257,283,304,327,350,380,416,443,471,516,557
%N A260221 Expansion of phi(x^3)^2 / f(x) in powers of x where phi(), f() are Ramanujan theta functions.
%C A260221 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260221 G. C. Greubel, <a href="/A260221/b260221.txt">Table of n, a(n) for n = 0..1000</a>
%H A260221 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260221 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260221 Expansion of f(-x, x^2)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%F A260221 Expansion of q^(1/24) * eta(q) * eta(q^4) * eta(q^6)^10 / (eta(q^2)^3 * eta(q^3)^4 * eta(q^12)^4) in powers of q.
%F A260221 Euler transform of period 12 sequence [ -1, 2, 3, 1, -1, -4, -1, 1, 3, 2, -1, -1, ...].
%F A260221 a(n) = A259538(3*n).
%e A260221 G.f. = 1 - x + 2*x^2 + x^3 + x^4 + x^5 + 3*x^6 + x^7 + 2*x^8 + 2*x^9 + ...
%e A260221 G.f. = 1/q - q^23 + 2*q^47 + q^71 + q^95 + q^119 + 3*q^143 + q^167 + ...
%t A260221 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] ^2 / QPochhammer[ -x], {x, 0, n}];
%o A260221 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^10 / (eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};
%Y A260221 Cf. A259538.
%K A260221 sign
%O A260221 0,3
%A A260221 _Michael Somos_, Jul 19 2015