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A263528 Expansion of (psi(x) * psi(x^3) / f(-x^3)^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

%I A263528 #10 Feb 16 2025 08:33:27
%S A263528 1,2,1,8,14,6,38,60,23,140,208,76,439,626,221,1232,1704,584,3182,4300,
%T A263528 1443,7700,10212,3368,17673,23076,7497,38808,50008,16046,82070,104560,
%U A263528 33190,167996,211920,66628,334202,417902,130288,648224,804254,248858,1229148
%N A263528 Expansion of (psi(x) * psi(x^3) / f(-x^3)^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.
%C A263528 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263528 G. C. Greubel, <a href="/A263528/b263528.txt">Table of n, a(n) for n = 0..2500</a>
%H A263528 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263528 Expansion of q^(-1/2) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^3)^3))^2 in powers of q.
%F A263528 Euler transform of period 6 sequence [ 2, -2, 8, -2, 2, 0, ...].
%F A263528 -2 * a(n) = A262930(2*n + 1).
%e A263528 G.f. = 1 + 2*x + x^2 + 8*x^3 + 14*x^4 + 6*x^5 + 38*x^6 + 60*x^7 + 23*x^8 + ...
%e A263528 G.f. = q + 2*q^3 + q^5 + 8*q^7 + 14*q^9 + 6*q^11 + 38*q^13 + 60*q^15 + 23*x^17 + ...
%t A263528 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)] / (4 QPochhammer[ x^3]^2))^2 / x, {x, 0, n}];
%o A263528 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A)^3))^2, n))};
%Y A263528 Cf. A262930.
%K A263528 nonn
%O A263528 0,2
%A A263528 _Michael Somos_, Oct 19 2015