%I #20 Mar 23 2026 17:05:02

%S 1,8,36,128,386,1024,2488,5632,12031,24576,48308,91904,170110,307200,

%T 542872,941056,1602819,2686976,4439688,7238272,11657090,18561024,

%U 29242240,45617664,70507772,108036096,164192188,247620352,370726652

%N McKay-Thompson series of class 8A for the Monster group with a(0) = 8.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. A. Edgar, <a href="/A131123/b131123.txt">Table of n, a(n) for n = -1..996</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], 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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^-1 * (chi(-q^4) / chi(-q))^8 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^4) / ( eta(q) * eta(q^8) ))^8 in powers of q.

%F Euler transform of period 8 sequence [ 8, 0, 8, -8, 8, 0, 8, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 16 * (1 - v*w) * (1 - v*u) - (v - u^2) * (v - w^2).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = A007265(n) unless n=0.

%F a(n) ~ exp(sqrt(2*n)*Pi) / (2^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%F Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * exp(-Pi) * sqrt(2) * (2+sqrt(2))^2 = A388592. - _Simon Plouffe_, Sep 18 2025

%e T8A = 1/q + 8 + 36*q + 128*q^2 + 386*q^3 + 1024*q^4 + 2488*q^5 + 5632*q^6 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^4, q^4])^8, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(8*k))))^8,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)))^8, n))};

%Y Cf. A007265, A045490.

%K nonn

%O -1,2

%A _Michael Somos_, Jun 15 2007