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A207735 Expansion of f(-x^2, x^3)^2 / f(x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

%I A207735 #14 Feb 16 2025 08:33:16
%S A207735 1,-1,0,1,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,-1,
%T A207735 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A207735 -1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A207735 Expansion of f(-x^2, x^3)^2 / f(x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
%C A207735 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A207735 This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = x.
%H A207735 G. C. Greubel, <a href="/A207735/b207735.txt">Table of n, a(n) for n = 0..1000</a>
%H A207735 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A207735 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A207735 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A207735 Expansion of f(x^7, -x^8) - x * f(-x^2, x^13) = f(x^5, -x^10) * f(-x^2, x^3) / f(x, -x^4) where f() is Ramanujan's two-variable theta function.
%F A207735 Euler transform of period 20 sequence [ -1, 0, 1, 1, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 1, 1, 0, -1, -1, ...].
%F A207735 G.f.: Sum_{k} (-1)^[k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) - x^(3*k + 1)).
%F A207735 |a(n)| is the characteristic function of A093722.
%F A207735 The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
%F A207735 a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113681(n).
%e A207735 1 - x + x^3 + x^7 - x^8 - x^14 + x^20 - x^29 - x^31 + x^42 - x^52 - x^66 + ...
%e A207735 q - q^121 + q^361 + q^841 - q^961 - q^1681 + q^2401 - q^3481 - q^3721 + ...
%t A207735 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207735[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[x, -x^4], {x, 0, n}]; Table[A207735[n], {n,0,50}] (* _G. C. Greubel_, Jun 18 2017 *)
%o A207735 (PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^n * kronecker( 12, m), 0)}
%Y A207735 Cf. A057538, A093722, A113681.
%K A207735 sign
%O A207735 0,1
%A A207735 _Michael Somos_, Feb 19 2012