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A328796 Expansion of chi(x) / chi(-x^6) in powers of x where chi() is a Ramanujan theta function.

%I A328796 #13 Feb 16 2025 08:33:58
%S A328796 1,1,0,1,1,1,2,2,2,3,3,3,5,5,5,7,8,8,11,12,12,16,17,18,23,25,26,32,35,
%T A328796 37,45,49,52,62,67,72,85,92,98,114,124,133,153,166,178,203,220,236,
%U A328796 268,290,311,350,379,407,456,493,529,589,636,683,758,818,877
%N A328796 Expansion of chi(x) / chi(-x^6) in powers of x where chi() is a Ramanujan theta function.
%C A328796 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328796 Convolution square is A328790.
%C A328796 G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A328880.
%H A328796 Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/2302.01253">6-regular partitions: new combinatorial properties, congruences, and linear inequalities</a>, arXiv:2302.01253 [math.NT], 2023.
%H A328796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328796 Expansion of q^(-5/24) * (eta(q^2)^2 * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in power of q.
%F A328796 Euler transform of period 12 sequence [1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, ...].
%F A328796 G.f.: Product_{k>=1} (1 + x^(6*k))/(1 + (-x)^k) = Product_{k>=1} (1 + x^(2*k-1)) * (1 + x^(6*k)).
%F A328796 A261736(n) = (-1)^n * a(n).
%F A328796 a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019
%e A328796 G.f. = 1 + x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
%e A328796 G.f. = q^5 + q^29 + q^77 + q^101 + q^125 + 2*q^149 + 2*q^173 + ...
%t A328796 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6], {x, 0, n}];
%o A328796 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
%Y A328796 Cf. A261736, A328790, A328800.
%K A328796 nonn
%O A328796 0,7
%A A328796 _Michael Somos_, Oct 27 2019