A258917 McKay-Thompson series of class 3A for the Monster group with a(0) = -66.

1, -66, 783, 8672, 65367, 371520, 1741655, 7161696, 26567946, 90521472, 288078201, 864924480, 2469235686, 6748494912, 17746495281, 45086909440, 111066966315, 266057139456, 621284327856, 1417338712800, 3164665156308, 6927097095040, 14885655834663

Offset

-1,2

Links

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

Formula

Expansion of ((eta(q) / eta(q^3))^6 - 27 * (eta(q^3) / eta(q))^6)^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 1 / f(t) where q = exp(2 Pi i t).

Convolution square of A007260.

a(n) = A007243(n) = A030197(n) = A045480(n) unless n = 0.

a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1/q - 66 + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/q) ((QPochhammer[ q] / QPochhammer[ q^3])^6 - 27 q (QPochhammer[ q^3] / QPochhammer[ q])^6)^2, {q, 0, n} ];

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^3 + A))^6 - 27 * x * (eta(x^3 + A) / eta(x + A))^6)^2, n))};

Crossrefs

Cf. A007243, A007260, A030197, A045480.

Sequence in context: A282050 A270258 A223410 * A223152 A301544 A229328

Adjacent sequences: A258914 A258915 A258916 * A258918 A258919 A258920

Keyword

sign

Author

Michael Somos, Jun 14 2015

Status

approved