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A230535 Expansion of q * (f(-q, -q^7) / f(-q^3, -q^5))^2 in powers of q where f(,) is Ramanujan's two-variable theta function.

%I A230535 #25 Feb 16 2025 08:33:20
%S A230535 1,-2,1,2,-4,4,-1,-6,11,-8,-1,12,-20,16,2,-22,34,-30,1,40,-64,52,-2,
%T A230535 -68,113,-88,-2,112,-180,144,2,-182,284,-228,4,286,-452,356,-4,-440,
%U A230535 698,-544,-5,668,-1044,816,6,-996,1545,-1210,6,1464,-2276,1768,-7,-2128
%N A230535 Expansion of q * (f(-q, -q^7) / f(-q^3, -q^5))^2 in powers of q where f(,) is Ramanujan's two-variable theta function.
%C A230535 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230535 G. C. Greubel, <a href="/A230535/b230535.txt">Table of n, a(n) for n = 1..1000</a>
%H A230535 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230535 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230535 Euler transform of period 8 sequence [ -2, 0, 2, 0, 2, 0, -2, 0, ...].
%F A230535 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - u^2) * (1 - v)  - 4*u*v.
%F A230535 G.f.: x * (Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))))^2.
%F A230535 a(2*n) = -2 * A224216(n). a(2*n + 1) = A230534(n).
%F A230535 Convolution square of A092869.
%F A230535 Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)). - _Simon Plouffe_, Mar 02 2021
%e A230535 G.f. = q - 2*q^2 + q^3 + 2*q^4 - 4*q^5 + 4*q^6 - q^7 - 6*q^8 + 11*q^9 + ...
%t A230535 a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 8, k], {k, n}]^2, {q, 0, n}];
%t A230535 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2] / (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]))^2, {q, 0, n}];
%t A230535 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^8] QPochhammer[ q^7, q^8] / (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8]))^2, {q, 0, n}];
%o A230535 (PARI) {a(n) = local(A, A2); if( n<2, n==1, n--; A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( (2 * A^2 * A2^2 / (A^2 + A2))^2, n))};
%Y A230535 Cf. A092869, A224216, A230534.
%K A230535 sign
%O A230535 1,2
%A A230535 _Michael Somos_, Oct 22 2013