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A328790 Expansion of (chi(x) / chi(-x^6))^2 in powers of x where chi() is a Ramanujan theta function.

%I A328790 #11 Feb 16 2025 08:33:58
%S A328790 1,2,1,2,4,4,7,10,11,16,21,24,34,44,50,66,84,98,125,156,181,226,277,
%T A328790 322,397,480,557,674,807,936,1121,1330,1538,1824,2146,2476,2915,3408,
%U A328790 3918,4578,5322,6104,7090,8198,9375,10830,12464,14214,16345,18734,21303
%N A328790 Expansion of (chi(x) / chi(-x^6))^2 in powers of x where chi() is a Ramanujan theta function.
%C A328790 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328790 Convolution square of A328796.
%C A328790 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328797.
%H A328790 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328790 Expansion of q^(-5/12) * (eta(q^2)^2 * eta(q^12))^2 / (eta(q) * eta(q^4) * eta(q^6))^2 in power of q.
%F A328790 Euler transform of period 12 sequence [2, -2, 2, 0, 2, 0, 2, 0, 2, -2, 2, 0, ...].
%F A328790 G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 + x^(6*k))^2.
%F A328790 a(n) = A112206(2*n + 1).
%F A328790 a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019
%e A328790 G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 10*x^7 + ...
%e A328790 G.f. = q^5 + 2*q^17 + q^29 + 2*q^41 + 4*q^53 + 4*q^65 + 7*q^77 + ...
%t A328790 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6])^2, {x, 0, n}];
%o A328790 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^12 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};
%Y A328790 Cf. A112206, A328796, A328797.
%K A328790 nonn
%O A328790 0,2
%A A328790 _Michael Somos_, Oct 27 2019