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A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two-variable theta function.

%I A207710 #14 Feb 16 2025 08:33:16
%S A207710 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 A207710 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,
%U A207710 0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two-variable theta function.
%C A207710 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A207710 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 A207710 G. C. Greubel, <a href="/A207710/b207710.txt">Table of n, a(n) for n = 0..1000</a>
%H A207710 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A207710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A207710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A207710 Expansion of f(x^5, -x^10) * f(-x^2, x^3) / f(-x, x^4) = f(-x^7, x^8) + x * f(x^2, -x^13) in powers of x where f() is Ramanujan's two-variable theta function.
%F A207710 Euler transform of period 20 sequence [ 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, ...].
%F A207710 |a(n)| is the characteristic function of A093722.
%F A207710 The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
%F A207710 G.f.: Prod_{k>0} (1 - (-x)^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))).
%F A207710 G.f.: Sum_{k} (-1)^[-k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
%F A207710 a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113430(n).
%e A207710 1 + x + x^3 - x^7 + x^8 - x^14 - x^20 - x^29 - x^31 - x^42 - x^52 + x^66 + ...
%e A207710 q + q^121 + q^361 - q^841 + q^961 - q^1681 - q^2401 - q^3481 - q^3721 + ...
%t A207710 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207710[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[-x, x^4], {x, 0, n}]; Table[A207710[n], {n,0,50}] (* _G. C. Greubel_, Jun 18 2017 *)
%o A207710 (PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), kronecker( -120, m) * (-1)^(m \ 15))}
%Y A207710 Cf. A057538, A093722, A113430.
%K A207710 sign
%O A207710 0,1
%A A207710 _Michael Somos_, Feb 19 2012