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%I #44 Sep 26 2023 09:58:13
%S 1,42,783,8672,65367,371520,1741655,7161696,26567946,90521472,
%T 288078201,864924480,2469235686,6748494912,17746495281,45086909440,
%U 111066966315,266057139456,621284327856,1417338712800,3164665156308
%N McKay-Thompson series of class 3A for the Monster group with a(0) = 42.
%C (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
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%D N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 39.
%D D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%D 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
%H Vaclav Kotesovec, <a href="/A030197/b030197.txt">Table of n, a(n) for n = -1..2000</a>
%H T. Piezas III, <a href="https://sites.google.com/view/tpiezas/0013-article-3-pi-formulas-and-the-monster">0013: Article 3 (Pi Formulas and the Monster Group)</a>
%H Titus Piezas III, <a href="http://www.oocities.org/titus_piezas/Pi_formulas2.pdf">On Ramanujan's Other Pi Formulas</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of Hauptmodul for X_0^{+}(3).
%F Expansion of (h + 27)^2 / h, where h = (eta(q) / eta(q^3))^12.
%F 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
%F 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
%F 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
%F a(n) ~ exp(4*Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2015
%e G.f. = 1/q + 42 + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...
%t 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 *)
%o (PARI) {a(n) = my(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 */
%Y Apart from constant term, same as A007243, A045480.
%Y Cf. A007261. - _Gary W. Adamson_, Jul 21 2009
%Y Cf. A058092, A058537, A121590.
%K nonn,nice,easy
%O -1,2
%A _N. J. A. Sloane_
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