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A007243 McKay-Thompson series of class 3A for the Monster group with a(0) = 0.
(Formerly M5480)
5
1, 0, 783, 8672, 65367, 371520, 1741655, 7161696, 26567946, 90521472, 288078201, 864924480, 2469235686, 6748494912, 17746495281, 45086909440, 111066966315, 266057139456, 621284327856, 1417338712800, 3164665156308 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

Expansion of Hauptmodul for X_0^{+}(3).

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

a(n) = A030197(n) = A045480(n) unless n = 0.

a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 01 2017

EXAMPLE

T3A = 1/q + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...

MATHEMATICA

QP = QPochhammer; A = q*O[q]^20; A = (QP[q^3+A]/QP[q+A])^12; s = (1+27*q* A)^2/A - 42*q; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 12 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( (1 + 27 * x * A)^2 / A - 42 * x, n))} /* Michael Somos, Feb 02 2012 */

CROSSREFS

Cf. A030197, A045480.

Sequence in context: A045074 A204279 A158399 * A146978 A095954 A257355

Adjacent sequences:  A007240 A007241 A007242 * A007244 A007245 A007246

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 21 17:07 EST 2019. Contains 319350 sequences. (Running on oeis4.)