A058548 McKay-Thompson series of class 18j for the Monster group.

1, 0, 1, -2, 0, 2, 1, 0, 3, 2, 0, 2, -4, 0, 1, 0, 0, 2, 7, 0, 4, -10, 0, 8, 3, 0, 8, 10, 0, 8, -15, 0, 7, 2, 0, 10, 22, 0, 17, -32, 0, 22, 10, 0, 26, 32, 0, 24, -48, 0, 25, 8, 0, 30, 62, 0, 43, -88, 0, 58, 22, 0, 65, 88, 0, 66, -127, 0, 66, 22, 0, 80, 152, 0, 107, -214, 0, 136, 52, 0

Offset

-1,4

Comments

G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) = f(A(x), A(-x)) where f(u, v) = 32 + 4 * (u + v) - 2 * (u^2 + v^2) + 2 * (u^3 + v^3) - 3*u*v * (u + v) + (u^4 + v^4) + u*v * (u^2 + v^2) - (u*v)^2 * (u + v). - Michael Somos, Apr 20 2004

Links

G. C. Greubel, Table of n, a(n) for n = -1..2500

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Index entries for McKay-Thompson series for Monster simple group

Formula

a(3*n) = 0.

Expansion of A + 1/A, where A = (eta(q^3)*eta(q^18)^2*eta(q^27)/(eta(q^6) *eta(q^9)^2*eta(q^54)))^2, in powers of q. - G. C. Greubel, Jun 21 2018

Example

T18j = 1/q + q - 2*q^2 + 2*q^4 + q^5 + 3*q^7 + 2*q^8 + 2*q^10 - 4*q^11 + ...

Mathematica

nmax = 80; QP = QPochhammer; A = x^2*O[x]^nmax; A = ((QP[A + x^3]*QP[A + x^18]^2*QP[A + x^27])/(QP[A + x^6]*QP[A + x^9]^2*QP[A + x^54]))^2/x; a[n_] := SeriesCoefficient[A + 1/A, n]; Table[a[n], {n, -1, nmax}] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; A := q*(eta[q^3]*eta[q^18]^2* eta[q^27]/( eta[q^6]*eta[q^9]^2*eta[q^54]))^2; a := CoefficientList[ Series[A + q^2/A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 21 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, n==-1, A = x^2 * O(x^n); A = ((eta(x^3 + A) * eta(x^18 + A)^2 * eta(x^27 + A)) / (eta(x^6 + A) * eta(x^9 + A)^2 * eta(x^54 + A)))^2 / x; polcoeff( A + 1/A, n))} /* Michael Somos, Apr 20 2004 */

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Sequence in context: A321460 A177446 A074905 * A372876 A157030 A080844

Adjacent sequences: A058545 A058546 A058547 * A058549 A058550 A058551

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved