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%I #13 Mar 12 2021 22:24:48
%S 1,2,4,8,8,12,16,16,29,36,44,64,72,88,112,128,162,202,244,304,352,420,
%T 496,576,703,820,968,1152,1320,1544,1792,2048,2405,2782,3204,3728,
%U 4240,4856,5568,6320,7259,8276,9416,10752,12144,13760,15568,17536,19875,22416
%N Expansion of phi(x) * chi(x^2)^4 in powers of x where phi(), chi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A260514/b260514.txt">Table of n, a(n) for n = 0..2500</a>
%H Vaclav Kotesovec, <a href="http://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], Sep 30 2015
%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 q^(1/3) * eta(q^2) * eta(q^4)^6 / (eta(q)^2 * eta(q^8)^4) in powers of q.
%F Euler transform of period 8 sequence [ 2, 1, 2, -5, 2, 1, 2, -1, ...].
%F a(n) ~ exp(Pi*sqrt(n/3)) / (2*sqrt(n)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 16*x^7 + ...
%e G.f. = 1/q + 2*q^2 + 4*q^5 + 8*q^8 + 8*q^11 + 12*q^14 + 16*q^17 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4]^4, {x, 0, n}];
%t nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(4*k))^6 / ((1-x^k) * (1-x^(8*k))^4),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^4)^6/(eta(q)^2*eta(q^8)^4)) \\ _Altug Alkan_, Aug 01 2018
%K nonn
%O 0,2
%A _Michael Somos_, Jul 27 2015
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