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A134781 McKay-Thompson series of class 23A for the Monster group with a(0) = 1. 2
1, 1, 4, 7, 13, 19, 33, 47, 74, 106, 154, 214, 307, 417, 575, 772, 1045, 1379, 1837, 2394, 3135, 4048, 5232, 6686, 8560, 10840, 13737, 17273, 21701, 27086, 33783, 41890, 51893, 63969, 78748, 96536, 118196, 144146, 175561, 213122, 258327, 312202 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

A058570(n) = a(n) unless n=0.

LINKS

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

M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Associated with permutations in Mathieu group M24 of shape (23)(1).

G.f. is Fourier series of a level 23 modular function. f(-1/ (23 t)) = f(t) where q = exp(2 Pi i t).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) + 2*(u*v * (u + v) + 2*(u^2 + v^2) + 5*u*v + 3*(u + v) + 1).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (2 - u - w) + v*(9 + 2*(u + w)) + u^2 + u*w + w^2 + 4*(u + w) + 6.

G.f.: (Sum_{j,k} x^(2*j^2 + j*k + 3*k^2)) / (x * Product_{k>0} (1 - x^k) * (1 - x^(23*k))).

a(n) ~ exp(4*Pi*sqrt(n/23)) / (sqrt(2) * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

EXAMPLE

1/q + 1 + 4*q + 7*q^2 + 13*q^3 + 19*q^4 + 33*q^5 + 47*q^6 + 74*q^7 + ...

MATHEMATICA

nmax = 40; QP = QPochhammer; s = Sum[x^(2*j^2 + j*k + 3*k^2), {j, -nmax, nmax}, {k, -nmax, nmax}]/(QP[x]*QP[x^23]) + O[x]^nmax; CoefficientList[s, x] (* Jean-Fran├žois Alcover, Nov 15 2015 *)

eta[q_] := q^(1/24)*QPochhammer[q]; e46A := (eta[q]*eta[q^23]/(eta[q^2] *eta[q^46])); T23A := (e46A + 1)*(e46A^2 + 4)/e46A^2;  Table[ SeriesCoefficient[1 + T23A, {q, 0, n}], {n, -1, 50}] (* G. C. Greubel, Feb 13 2018 *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (1 + 2 * x * Ser(qfrep([4, 1; 1, 6], n, 1))) / (eta(x + A) * eta(x^23 + A)), n))}

CROSSREFS

Convolution with A030199 is A028930.

Sequence in context: A216880 A144730 A058570 * A127977 A100848 A051458

Adjacent sequences:  A134778 A134779 A134780 * A134782 A134783 A134784

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 12 2007

STATUS

approved

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Last modified January 18 03:13 EST 2019. Contains 319260 sequences. (Running on oeis4.)