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%I #21 Mar 12 2021 22:24:47
%S 1,1,1,1,1,2,4,5,5,5,6,9,12,15,16,17,20,26,34,40,44,48,55,68,84,98,
%T 108,118,135,161,192,221,244,268,303,354,414,470,519,571,641,737,847,
%U 954,1052,1156,1291,1465,1664,1861,2048,2248,2496,2807,3158,3511,3855
%N Expansion of chi(x^6) / (chi(-x) * chi(x^3)) in powers of x which chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A227401/b227401.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(x^6) / f(-x^1, -x^5) in powers of x where f(,) is a Ramanujan theta function.
%F Expansion of q^(1/12) * eta(q^2) * eta(q^3) * eta(q^12)^3 / (eta(q) * eta(q^6)^3 * eta(q^24)) in powers of q.
%F Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 0, ...].
%F a(n) ~ 11^(1/4) * exp(Pi*sqrt(11*n)/6) / (4*sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, Jul 11 2016
%e G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 5*x^8 + 5*x^9 + ...
%e G.f. = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 4*q^71 + 5*q^83 + 5*q^95 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^6] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]), {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^6, x^12] QPochhammer[ -x, x] QPochhammer[ x^3, -x^3], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^6 + A)^3 * eta(x^24 + A)), n))};
%K nonn
%O 0,6
%A _Michael Somos_, Sep 20 2013
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