A215660 McKay-Thompson series of class 18B for the Monster group with a(0) = 5.

1, 5, 7, 10, 27, 38, 82, 108, 207, 278, 486, 644, 1052, 1404, 2182, 2880, 4293, 5654, 8182, 10692, 15076, 19604, 27108, 35000, 47547, 61020, 81713, 104236, 137781, 174800, 228498, 288360, 373174, 468566, 601020, 751036, 955642, 1188756, 1501730, 1859944

Offset

-1,2

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Formula

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.

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A215413(n) + 4 * A212484(n).

a(n) = A058532(n) = A215407(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

1/q + 5 + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...

Mathematica

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 *)
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 *)

Prog

(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))}

Crossrefs

Cf. A058532, A212484, A215407, A215413.

Sequence in context: A088768 A131998 A141443 * A015849 A059303 A061523

Adjacent sequences: A215657 A215658 A215659 * A215661 A215662 A215663

Keyword

nonn

Author

Michael Somos, Aug 19 2012

Status

approved