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A225701 Expansion of chi(q)^5 / chi(q^5) in powers of q where chi() is a Ramanujan theta function.

%I A225701 #37 Feb 16 2025 08:33:19
%S A225701 1,5,10,15,30,55,80,120,190,285,410,585,840,1190,1640,2240,3070,4170,
%T A225701 5570,7400,9830,12960,16920,21990,28520,36805,47180,60225,76720,97350,
%U A225701 122880,154610,194110,242880,302740,376295,466710,577270,711800,875520,1074790
%N A225701 Expansion of chi(q)^5 / chi(q^5) in powers of q where chi() is a Ramanujan theta function.
%C A225701 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A225701 G. C. Greubel, <a href="/A225701/b225701.txt">Table of n, a(n) for n = 0..1000</a>
%H A225701 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], 2015-2016.
%H A225701 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A225701 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A225701 Expansion of eta(q^2)^10 * eta(q^5) * eta(q^20) / (eta(q)^5 * eta(q^4)^5 * eta(q^10)^2) in powers of q.
%F A225701 Euler transform of period 20 sequence [ 5, -5, 5, 0, 4, -5, 5, 0, 5, -4, 5, 0, 5, -5, 4, 0, 5, -5, 5, 0, ...].
%F A225701 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A223903.
%F A225701 a(n) = (-1)^n * A138521(n). a(n) = 5 * A210458(n) unless n=0.
%F A225701 a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%F A225701 Empirical: Sum_{n>=0} a(n)/exp(Pi*n) = sqrt(5) - 1. - _Simon Plouffe_, Mar 02 2021
%e A225701 G.f. = 1 + 5*q + 10*q^2 + 15*q^3 + 30*q^4 + 55*q^5 + 80*q^6 + 120*q^7 + 190*q^8 + ...
%t A225701 a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^5 / QPochhammer[ -q^5, q^10], {q, 0, n}];
%t A225701 nmax=60; CoefficientList[Series[Product[(1-x^k)^5 * (1+x^k)^10 * (1+x^(10*k)) / ((1-x^(4*k))^5 * (1+x^(5*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o A225701 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^10 + A)^2), n))};
%Y A225701 Cf. A138521, A210458, A223903.
%K A225701 nonn
%O A225701 0,2
%A A225701 _Michael Somos_, May 17 2013