A035099 McKay-Thompson series of class 2B for the Monster group with a(0) = 40.

1, 40, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144

Offset

-1,2

Comments

Also Fourier coefficients of j_2 where j_2 is an analytic isomorphism H/\Gamma_0(2) ->\hat{C}.

"The function j_2 is analogous to j because it is modular (weight zero) for \Gamma_0(2), holomorphic on the upper half-plane, has a simple pole at infinity, generates the field of \Gamma_0(2)-modular functions, and defines a bijection of a \Gamma_0(2) fundamental set with C." from the Brent article page 260 using his notation of j_2. - Michael Somos, Mar 08 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

Links

T. D. Noe, Table of n, a(n) for n=-1..1000

R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274; see also Project Euclid

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

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

G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Monster Group

Index entries for "core" sequences

Formula

Expansion of 64 + q^(-1) * (phi(-q) / psi(q))^8 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 08 2011

Expansion of 64 + (eta(q) / eta(q^2))^24 in powers of q. - Michael Somos, Mar 08 2011

j_2 = E_{gamma, 2}^2 / E_{oo, 4} in the notation of Brent where E_{gamma, 2} is g.f. for A004011 and E_{oo, 4} is g.f. for A007331. - Michael Somos, Mar 08 2011

G.f.: 64 + x^(-1) * (Product_{k>0} 1 + x^k)^(-24). - Michael Somos, Mar 08 2011

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Nov 16 2016

Example

j_2 = 1/q + 40 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...

Mathematica

max = 21; f[x_] := Product[ 1 + x^k, {k, 1, max}]^(-24); coes = CoefficientList[ Series[ f[x], {x, 0, max} ], x]; a[n_] := coes[[n+2]]; a[0] = 40; Table[a[n], {n, -1, max-1}] (* Jean-François Alcover, Nov 03 2011, after Michael Somos *)
QP = QPochhammer; s = 64*q + (QP[q]/QP[q^2])^24 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, after Michael Somos *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 64 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Mar 08 2011 */

Crossrefs

Cf. A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.

Sequence in context: A229588 A334121 A117216 * A065255 A300920 A061993

Adjacent sequences: A035096 A035097 A035098 * A035100 A035101 A035102

Keyword

easy,sign,nice,core

Author

Barry Brent (barryb(AT)primenet.com)

Status

approved