A030183 McKay-Thompson series of class 7A for the Monster group with a(0) = 10.

1, 10, 51, 204, 681, 1956, 5135, 12360, 28119, 60572, 125682, 251040, 487426, 920568, 1699611, 3070508, 5445510, 9490116, 16283793, 27537708, 45959775, 75760640, 123471327, 199081632, 317814988

Offset

-1,2

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

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

N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 39.

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

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.

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of Hauptmodul for X_0^{+}(7).

Expansion of (h + 7)^2 / h, where h = (eta(q) / eta(q^7))^4 in powers of q.

a(n) = A007264(n) = A045489(n) unless n = 0.

a(n) ~ exp(4*Pi*sqrt(n/7)) / (sqrt(2) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = 1/q + 10 + 51*q + 204*q^2 + 681*q^3 + 1956*q^4 + 5135*q^5 + 12360*q^6 + ...

Mathematica

a[ n_] := With[ {A = q (QPochhammer[ q^7] / QPochhammer[ q])^4}, SeriesCoefficient[ (1 + 7 A)^2 / A, {q, 0, n}]]; (* Michael Somos, Mar 30 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1 + 7 * x * A)^2 / A, n))}; /* Michael Somos, Feb 02 2012 */

Crossrefs

Cf. A007264, A045489.

Sequence in context: A077044 A069038 A213563 * A224327 A219573 A135242

Adjacent sequences: A030180 A030181 A030182 * A030184 A030185 A030186

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved