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A229180 Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function.

%I A229180 #21 Feb 16 2025 08:33:20
%S A229180 1,3,6,16,33,60,118,210,354,612,1008,1608,2583,4035,6174,9448,14196,
%T A229180 21024,31054,45282,65322,93884,133638,188640,265225,370086,512934,
%U A229180 708136,971628,1325724,1802134,2437200,3280452,4400132,5876184,7815288,10360890,13683525
%N A229180 Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function.
%C A229180 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A229180 In Verrill (1999) section 2.6, denoted by g as a function of q.
%D A229180 H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.
%H A229180 G. C. Greubel, <a href="/A229180/b229180.txt">Table of n, a(n) for n = 0..2500</a>
%H A229180 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A229180 H. Verrill, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=46">Some Congruences related to modular forms</a>
%H A229180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A229180 Expansion of q^(-1/2) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^3 in powers of q.
%F A229180 Euler transform of period 6 sequence [3, 0, 6, 0, 3, 0, ...].
%F A229180 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058492.
%F A229180 G.f.: t / (1 - 10*t^2 + 9*t^4)^(1/2) where t = the g.f. of A217786.
%F A229180 G.f.: 1 / (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3.
%F A229180 Convolution inverse of A058492.
%F A229180 a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e A229180 G.f. = 1 + 3*x + 6*x^2 + 16*x^3 + 33*x^4 + 60*x^5 + 118*x^6 + 210*x^7 + ...
%e A229180 G.f. = q + 3*q^3 + 6*q^5 + 16*q^7 + 33*q^9 + 60*q^11 + 118*q^13 + 210*q^15 + ...
%t A229180 a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[x^3, x^6])^3, {x, 0, n}];
%t A229180 nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%o A229180 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^3, n))};
%Y A229180 Cf. A058492, A084471, A217786.
%K A229180 nonn
%O A229180 0,2
%A A229180 _Michael Somos_, Sep 30 2013