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A030182
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McKay-Thompson series of class 3B for the Monster group with a(0) = -12.
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5
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1, -12, 54, -76, -243, 1188, -1384, -2916, 11934, -11580, -21870, 79704, -71022, -123444, 421308, -352544, -581013, 1885572, -1510236, -2388204, 7469928, -5777672, -8852004, 26869968, -20218587, -30177684, 89408826
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OFFSET
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-1,2
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COMMENTS
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Let t(q) = (eta(q)/eta(q^3))^12 = 1/q-12+54q-76q^2-243q^3+.... If j(q) is the j-invariant, with q-series given by A000521, then j(q) = (t+27)(t+243)^3/t^3 j(q^3) = (t+27)(t+3)^3/t. Hence t(q) can be used to parametrize the classical modular curve X0(3). - Gene Ward Smith (genewardsmith(AT)gmail.com), Aug 04 2006
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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. 38.
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.
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LINKS
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Table of n, a(n) for n=-1..25.
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
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Expansion of (eta(q) / eta(q^3))^12 in powers of q.
Expansion of (3 * b(q) / c(q))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Euler transform of period 3 sequence [ -12, -12, 0, ...]. - Michael Somos, Nov 08 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^3 - u * (27 + u) * v * (27 + v). - Michael Somos, Nov 08 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 / f(t) where q = exp(2 pi i t). - Michael Somos, Nov 08 2011
G.f.: x^-1 * (Product_{k>0} (1 - x^k) / (1 - x^(3*k)))^12.
Convolution inverse of A121590. Convolution square of A007262. Convolution cube of A058095. Convolution fourth power of A199659. Convolution sixth power of A112157. Convolution twelfth power of A137569.
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EXAMPLE
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1/q - 12 + 54*q - 76*q^2 - 243*q^3 + 1188*q^4 - 1384*q^5 - 2916*q^6 + ...
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MATHEMATICA
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a[ n_] := With[{m = n + 1}, SeriesCoefficient[ (Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 3, m, 3}])^12, {q, 0, m}]] (* Michael Somos, Nov 08 2011 *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^12, n))} /* Michael Somos, Nov 08 2011 */
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CROSSREFS
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Cf. A007244, A007262, A045481, A058095, A112157, A121590, A137569, A198955, A199659.
Sequence in context: A195544 A213549 A211060 * A060171 A133078 A034436
Adjacent sequences: A030179 A030180 A030181 * A030183 A030184 A030185
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KEYWORD
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sign,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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