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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
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
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.
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-François 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
Sequence in context: A045074 A204279 A158399 * A146978 A095954 A351476
KEYWORD
nonn,nice,easy
STATUS
approved