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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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6
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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, Aug 04 2006
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LINKS
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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(q) satisfies 0 = f(A(q), A(q^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 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121590. - Michael Somos, Nov 08 2011
G.f.: x^-1 * (Product_{k>0} (1 - x^k) / (1 - x^(3*k)))^12.
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EXAMPLE
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G.f. = 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 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^3])^12, {q, 0, n}]; (* Michael Somos, May 03 2015 *)
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PROG
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(PARI) {a(n) = my(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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KEYWORD
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sign,nice,easy
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AUTHOR
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STATUS
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approved
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