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A105559
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McKay-Thompson series of class 6E for the Monster group with a(0) = 3.
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8
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1, 3, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744, -20448, -46944
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OFFSET
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-1,2
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COMMENTS
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Number 2 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(6). [Yang 2004] - Michael Somos, Jul 21 2014
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LINKS
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FORMULA
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Expansion of (eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3))^3 in powers of q.
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = v^2 + 8*u + 6*u*v - u^2*v.
G.f.: x^-1 (Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k - 1)))^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128643.
Expansion of (c(q) / c(q^2))^3 in powers of q where c() is a cubic AGM theta function.
Expansion of q^(-1) * (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ 3, 0, -6, 0, 3, 0, ...].
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EXAMPLE
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G.f. = 1/q + 3 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 3, n, 6}] / Product[ 1 - q^k, {k, 1, n, 2}]^3)^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
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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^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3) )^3, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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