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Expansion of (chi(x) / chi(-x^6))^2 in powers of x where chi() is a Ramanujan theta function.
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%I #10 Oct 31 2019 17:33:45

%S 1,2,1,2,4,4,7,10,11,16,21,24,34,44,50,66,84,98,125,156,181,226,277,

%T 322,397,480,557,674,807,936,1121,1330,1538,1824,2146,2476,2915,3408,

%U 3918,4578,5322,6104,7090,8198,9375,10830,12464,14214,16345,18734,21303

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Convolution square of A328796.

%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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 Euler transform of period 12 sequence [2, -2, 2, 0, 2, 0, 2, 0, 2, -2, 2, 0, ...].

%F G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 + x^(6*k))^2.

%F a(n) = A112206(2*n + 1).

%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019

%e G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 10*x^7 + ...

%e G.f. = q^5 + 2*q^17 + q^29 + 2*q^41 + 4*q^53 + 4*q^65 + 7*q^77 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6])^2, {x, 0, n}];

%o (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 Cf. A112206, A328796, A328797.

%K nonn

%O 0,2

%A _Michael Somos_, Oct 27 2019