|
%I #15 Jul 03 2018 13:42:53
%S 1,5,7,10,27,38,82,108,207,278,486,644,1052,1404,2182,2880,4293,5654,
%T 8182,10692,15076,19604,27108,35000,47547,61020,81713,104236,137781,
%U 174800,228498,288360,373174,468566,601020,751036,955642,1188756,1501730,1859944
%N McKay-Thompson series of class 18B for the Monster group with a(0) = 5.
%H G. C. Greubel, <a href="/A215660/b215660.txt">Table of n, a(n) for n = -1..1000</a>
%F Expansion of (eta(q^3)^8 + 4 * eta(q^6)^8) / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^6)^2 * eta(q^9) * eta(q^18)) in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A215413(n) + 4 * A212484(n).
%F a(n) = A058532(n) = A215407(n) unless n=0.
%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e 1/q + 5 + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...
%t QP = QPochhammer; s = (QP[q^3]^8+4*q*QP[q^6]^8)/(QP[q]*QP[q^2]*QP[q^3]^2* QP[q^6]^2*QP[q^9]*QP[q^18]) + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[ q*(eta[q^3]^8 + 4*eta[q^6]^8)/(eta[q]*eta[q^2]*eta[q^3]^2*eta[q^6]^2* eta[q^9]*eta[q^18]), {q, 0, 100}], q]; Table[a[[n]], {n, 1, 80}] (* _G. C. Greubel_, Jul 03 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^18 + A)), n))}
%Y Cf. A058532, A212484, A215407, A215413.
%K nonn
%O -1,2
%A _Michael Somos_, Aug 19 2012
|
