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A259538 Expansion of psi(-x)^2 / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

%I A259538 #10 Feb 16 2025 08:33:26
%S A259538 1,-2,1,-1,0,1,2,-2,1,1,-2,0,1,-4,1,1,-2,2,3,-2,2,1,-4,2,2,-6,3,2,-4,
%T A259538 3,2,-6,3,4,-8,3,5,-10,5,3,-10,6,7,-10,5,8,-12,6,7,-16,8,7,-16,9,9,
%U A259538 -18,10,10,-22,10,11,-26,13,12,-26,15,14,-28,15,17,-34
%N A259538 Expansion of psi(-x)^2 / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.
%C A259538 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A259538 G. C. Greubel, <a href="/A259538/b259538.txt">Table of n, a(n) for n = 0..2500</a>
%H A259538 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259538 Expansion of q^(1/8) * eta(q)^2 * eta(q^4)^2 * eta(q^6) / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
%F A259538 Euler transform of period 12 sequence [ -2, 0, -1, -2, -2, 0, -2, -2, -1, 0, -2, -1, ...].
%F A259538 G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259529.
%F A259538 G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k)) / ((1 + x^k + x^(2*k)) * (1 - x^(2*k) + x^(4*k))).
%e A259538 G.f. = 1 - 2*x + x^2 - x^3 + x^5 + 2*x^6 - 2*x^7 + x^8 + x^9 - 2*x^10 + ...
%e A259538 G.f. = 1/q - 2*q^7 + q^15 - q^23 + q^39 + 2*q^47 - 2*q^55 + q^63 + ...
%t A259538 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2, 1}[[Mod[k, 12, 1]]], {k, n}], {x, 0, n}];
%t A259538 a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^4])^2 / ( QPochhammer[ x^3, x^6] QPochhammer[ x^12]), {x, 0, n}];
%o A259538 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2][k%12 + 1]), n))};
%o A259538 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y A259538 Cf. A259529.
%K A259538 sign
%O A259538 0,2
%A A259538 _Michael Somos_, Jun 30 2015