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 A030197 McKay-Thompson series of class 3A for the Monster group with a(0) = 42. 8
 1, 42, 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 -1,2 COMMENTS (1 + 42x + 783x^2 + 8672x^3 + ...) is the convolution square of (1 + 21x + 171x^2 + 745x^3 + ...), where A007261 = (1, 21, 171, 745, 2418,...).) -  Gary W. Adamson, Jul 21 2009 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). REFERENCES J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. N. D. Elkies, Elliptic and modular curves..., 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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = -1..2000 T. Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group) Titus Piezas III, On Ramanujan's Other Pi Formulas FORMULA Expansion of Hauptmodul for X_0^{+}(3). Expansion of (h + 27)^2 / h, where h = (eta(q) / eta(q^3))^12. Expansion of 27 * (b(q)^3 + c(q)^3)^2 / (b(q) * c(q))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012 Expansion of (a(q) / (eta(q) * eta(q^3)))^6 in powers of q where a() is a cubic AGM theta function. - Michael Somos, Dec 01 2013 G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 01 2013 a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015 EXAMPLE G.f. = 1/q + 42 + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 1/q ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / ( QPochhammer[ q] QPochhammer[ q^3]^2))^6, {q, 0, n}] (* Michael Somos, Dec 01 2013 *) 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, n))} /* Michael Somos, Jun 16 2012 */ CROSSREFS Apart from constant term, same as A007243, A045480. Cf. A007261 [From Gary W. Adamson, Jul 21 2009] Cf. A058092, A058537, A121590. Sequence in context: A159947 A252827 A231158 * A225980 A231164 A020933 Adjacent sequences:  A030194 A030195 A030196 * A030198 A030199 A030200 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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