A113298 - OEIS

%I #20 Mar 12 2021 22:24:43

%S 1,0,1,0,2,2,3,2,5,4,7,6,11,10,15,14,22,22,30,30,44,44,58,60,81,84,

%T 107,112,145,154,190,202,253,270,327,352,429,462,550,594,711,770,904,

%U 980,1156,1256,1457,1586,1845,2008,2310,2516,2898,3160,3604,3930,4488,4894

%N Expansion of q^(1/12) * eta(q^10)^5 / ( eta(q^2) * eta(q^5)^2 * eta(q^20)^2) in powers of q.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A113298/b113298.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 Euler transform of period 20 sequence [0, 1, 0, 1, 2, 1, 0, 1, 0, -2, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, ...].

%F Expansion of phi(q^5) / f(-q^2) in powers of q where phi(), f() are Ramanujan theta functions.

%F Expansion of G(x) * G(x^4) - x * H(x) * H(x^4) where G() = g.f. A003114 and H() = g.f. A003106 are the Rogers-Ramanujan functions.

%F G.f.: (1 + 2 * Sum_{k>0} x^(5*k^2)) / (Product_{k>0} (1 - x^(2*k))).

%F a(n) ~ exp(Pi*sqrt(n/3)) / (2*3^(1/4)*sqrt(10)*n^(3/4)). - _Vaclav Kotesovec_, Jul 11 2016

%e 1/q + q^23 + 2*q^47 + 2*q^59 + 3*q^71 + 2*q^83 + 5*q^95 + 4*q^107 + ...

%t nmax = 50; CoefficientList[Series[Product[(1-x^(10*k))^5 / ( (1-x^(2*k)) * (1-x^(5*k))^2 * (1-x^(20*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 11 2016 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(1/12)* eta[q^10]^5/(eta[q^2]*eta[q^5]^2*eta[q^20]^2), {q, 0, 50}], q] (* _G. C. Greubel_, Apr 17 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^5 / eta(x^2 + A) / eta(x^5 + A)^2 / eta(x^20 + A)^2, n))}

%o (PARI) q='q+O('q^99); Vec(eta(q^10)^5/(eta(q^2)*eta(q^5)^2*eta(q^20)^2)) \\ _Altug Alkan_, Apr 18 2018

%K nonn

%O 0,5

%A _Michael Somos_, Oct 24 2005