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A210458 Expansion of q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function.

%I A210458 #29 Feb 16 2025 08:33:17
%S A210458 1,2,3,6,11,16,24,38,57,82,117,168,238,328,448,614,834,1114,1480,1966,
%T A210458 2592,3384,4398,5704,7361,9436,12045,15344,19470,24576,30922,38822,
%U A210458 48576,60548,75259,93342,115454,142360,175104,214958,263262,321584,391993,476952
%N A210458 Expansion of q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function.
%C A210458 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210458 G. C. Greubel, <a href="/A210458/b210458.txt">Table of n, a(n) for n = 1..1000</a>
%H A210458 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 A210458 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210458 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210458 Expansion of (eta(q^2) * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10)))^2 in powers of q.
%F A210458 Euler transform of period 20 sequence [ 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, ...].
%F A210458 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 + u) * (1 + 5*u) * v * (1 + v) * (1 + 5*v).
%F A210458 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. of A145740.
%F A210458 G.f.: x * (Product_{k>0} P(5, x^k) * P(20, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.
%F A210458 A138519(n) = -(-1)^n * a(n). Convolution inverse of A145740.
%F A210458 a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%F A210458 Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -2/5 + (1/5)*sqrt(5). - _Simon Plouffe_, Mar 02 2021
%e A210458 q + 2*q^2 + 3*q^3 + 6*q^4 + 11*q^5 + 16*q^6 + 24*q^7 + 38*q^8 + 57*q^9 + ...
%t A210458 nmax=60; CoefficientList[Series[Product[((1+x^k) * (1-x^(5*k)) * (1+x^(10*k)) / (1-x^(4*k)))^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%t A210458 a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, I*q^(5/2)]/EllipticTheta[ 2, 0, I*Sqrt[q]])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Dec 07 2017 *)
%o A210458 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)))^2, n))}
%Y A210458 Cf. A138519, A145740.
%K A210458 nonn
%O A210458 1,2
%A A210458 _Michael Somos_, Jan 21 2013