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A030197
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McKay-Thompson series of class 3A for the Monster group with a(0) = 4.
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6
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1, 42, 783, 8672, 65367, 371520, 1741655, 7161696, 26567946, 90521472, 288078201, 864924480, 2469235686, 6748494912, 17746495281, 45086909440, 111066966315, 266057139456, 621284327856, 1417338712800, 3164665156308
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OFFSET
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-1,2
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COMMENTS
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(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
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REFERENCES
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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
Titus Piezas III, On Ramanujan's Other Pi Formulas, http://www.oocities.org/titus_piezas/Pi_formulas2.pdf
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LINKS
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Table of n, a(n) for n=-1..19.
Index entries for McKay-Thompson series for Monster simple group
T. Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group)
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FORMULA
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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
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EXAMPLE
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1/q + 42 + 783*q + 8672*q^2 + 65367*q^3 + 371520*q^4 + 1741655*q^5 + ...
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PROG
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(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 */
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CROSSREFS
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Apart from constant term, same as A007243, A045480.
Cf. A007261 [From Gary W. Adamson, Jul 21 2009]
Cf. A058092, A058537.
Sequence in context: A200853 A214945 A159947 * A225980 A020933 A030020
Adjacent sequences: A030194 A030195 A030196 * A030198 A030199 A030200
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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