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A208546 Expansion of f(-x^29, x^31) + x * f(-x^19, x^41) - x^3 * f(-x^11, x^49) + x^7 * f(x, -x^59) in powers of x where f() is Ramanujan's two-variable theta function.

%I A208546 #18 Feb 16 2025 08:33:16
%S A208546 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,-1,0,1,0,
%T A208546 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 A208546 -1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A208546 Expansion of f(-x^29, x^31) + x * f(-x^19, x^41) - x^3 * f(-x^11, x^49) + x^7 * f(x, -x^59) in powers of x where f() is Ramanujan's two-variable theta function.
%H A208546 G. C. Greubel, <a href="/A208546/b208546.txt">Table of n, a(n) for n = 0..2500</a>
%H A208546 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208546 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208546 |a(n)| is the characteristic function of A093722.
%F A208546 The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
%F A208546 Euler transform of a period 80 sequence.
%F A208546 G.f.: Sum_{k} (-1)^floor(k/4) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).
%e A208546 G.f. = 1 + x - x^3 + x^7 + x^8 + x^14 - x^20 - x^29 + x^31 + x^42 - x^52 - x^66 + ...
%e A208546 G.f. = q + q^121 - q^361 + q^841 + q^961 + q^1681 - q^2401 - q^3481 + q^3721 + ...
%t A208546 f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-x^29, x^31] + x*f[-x^19, x^41] - x^3*f[-x^11, x^49] + x^7*f[x, -x^59], {x, 0, 50}], x] (* _G. C. Greubel_, Aug 11 2018 *)
%o A208546 (PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 12))}
%Y A208546 Cf. A057538, A093722.
%K A208546 sign
%O A208546 0,1
%A A208546 _Michael Somos_, Feb 28 2012