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A007248
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McKay-Thompson series of class 4C for the Monster group.
(Formerly M5084)
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6
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1, 20, -62, 216, -641, 1636, -3778, 8248, -17277, 34664, -66878, 125312, -229252, 409676, -716420, 1230328, -2079227, 3460416, -5677816, 9198424, -14729608, 23328520, -36567242, 56774712, -87369461, 133321908, -201825396, 303248408, -452431503
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OFFSET
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0,2
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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.
D. Ford, J. McKay and S. P. Norton, ``More on replicable functions,'' Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
McKay, John; Strauss, Hubertus. The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..28.
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
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16*(theta_3/theta_2)^4 - 8 = 16 / lambda(z) - 8.
Expansion of q * ( -8 + 16 / lambda(z)) in powers of q^2 where nome q = exp(pi*i*z). - Michael Somos, Nov 14 2006
Expansion of q * (8 + (eta(q) / eta(q^4))^8) in powers of q^2. - Michael Somos, Nov 14 2006
Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (v + 24)^2 - (v + 8) * u^2 . - Michael Somos, Nov 14 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = g(t) where q = exp(2 pi i t) and g() is the g.f. for A097243. - Michael Somos, Jul 22 2011
A029845(2*n - 1) = A124972(2*n - 1) = a(n). - Michael Somos, Nov 14 2006.
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EXAMPLE
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T4C = 1/q + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 8 q + (QPochhammer[ q, q] / QPochhammer[ q^4, q^4])^8 , {q, 0, 2 n}] (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ -8 + 16 / m, {q, 0, 2 n - 1}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[ -8 + 16 (EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q])^4, {q, 0, 2 n - 1}] (* Michael Somos, Jul 22 2011 *)
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PROG
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(PARI) 8*x + prod(n=1, 39, if( n%4, 1 - x^n, 1), 1 + O(x^40))^8
(PARI) {a(n) = local(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8*x + (eta(x + A) / eta(x^4 + A))^8, n))} /* Michael Somos, Nov 14 2006 */
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CROSSREFS
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Cf. A029845, A124972.
Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
Sequence in context: A041784 A105092 A112144 * A117431 A159504 A117432
Adjacent sequences: A007245 A007246 A007247 * A007249 A007250 A007251
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KEYWORD
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sign,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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