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A058724 McKay-Thompson series of class 59A for the Monster group. 1
1, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 10, 10, 13, 15, 18, 20, 25, 28, 34, 38, 45, 50, 60, 67, 78, 88, 102, 114, 132, 147, 169, 189, 215, 240, 274, 304, 344, 383, 432, 479, 540, 597, 670, 742, 829, 916, 1023, 1128, 1255, 1384, 1536, 1690, 1874, 2059, 2277, 2501 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,5

COMMENTS

Also McKay-Thompson series of class 59B for the Monster group. - Michel Marcus, Feb 24 2014

The Monster conjugacy classes 59A and 59B are algebraic conjugates and so yield identical McKay-Thompson series. - Michael Somos, Jul 05 2014

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = -1..10000 (terms -1..2500 from G. C. Greubel)

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

David A. Madore, Coefficients of Moonshine (McKay-Thompson) series, The Math Forum

Index entries for McKay-Thompson series for Monster simple group

FORMULA

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) + 2*(u^2 + v^2) + 2*u*v + 2*(u + v) + 2. - Michael Somos, Jul 05 2014

Expansion of -1 + (G(q^59)*G(q) + q^12*H(q^59)*H(q))/q in powers of q, where G() is g.f. of A003114 and H() is g.f. of A003106. - G. C. Greubel, Jun 29 2018

a(n) ~ exp(4*Pi*sqrt(n/59)) / (sqrt(2) * 59^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018

EXAMPLE

T59A = 1/q + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...

MATHEMATICA

QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; G[x_] := f[-x^2, -x^3]/f[-x, -x^2]; H[x_] := f[-x, -x^4]/f[-x, -x^2]; A:= G[x^59]*G[x^1] + x^12*H[x^59]*H[x^1]; a:= CoefficientList[Series[A, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 29 2018 *)

PROG

(PARI) {a(n) = my(Q1, Q2); if( n<-1, 0, Q1 = 1 + 2*x * Ser( Vec( qfrep( [2, 1; 1, 30], n+2, 1))); Q2 = 1 + 2*x * Ser( Vec( qfrep( [6, 1; 1, 10], n+2, 1))); polcoeff( 2 / ( Q1/Q2 - 1), n))}; /* Michael Somos, Jul 05 2014 */

CROSSREFS

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

Sequence in context: A125894 A091493 A305148 * A029021 A261770 A096792

Adjacent sequences:  A058721 A058722 A058723 * A058725 A058726 A058727

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

EXTENSIONS

More terms from Michel Marcus, Feb 24 2014

STATUS

approved

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Last modified January 18 21:54 EST 2019. Contains 319282 sequences. (Running on oeis4.)