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A007247
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McKay-Thompson series of class 4B for the Monster group.
(Formerly M5305)
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2
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1, 52, 834, 4760, 24703, 94980, 343998, 1077496, 3222915, 8844712, 23381058, 58359168, 141244796, 327974700, 742169724, 1627202744, 3490345477, 7301071680, 14987511560, 30138820888, 59623576440, 115928963656
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| 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).
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
| Index entries for McKay-Thompson series for Monster simple group
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FORMULA
| Expansion of 4 * q * (1 + k'^2)^2 / (k' * k^2) in powers of q^2 where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
Expansion of 4 * q^(1/2) * (k'^4 + 4*k^2) / (k'^2 * k) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = f(t) where q = exp(2*pi*i*t). - Michael Somos, Jul 22 2011
a(n) = A007249(n) + 64 * A022577(n - 1). - Michael Somos, Jul 22 2011
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EXAMPLE
| T4B = 1/q + 52*q + 834*q^3 + 4760*q^5 + 24703*q^7 + 94980*q^9 + ...
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MATHEMATICA
| a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e + 64 / e), {q, 0, n - 1/2}]] (* Michael Somos Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 (2 - m)^2 / (m (1 - m)^(1/2)), {q, 0, 2 n - 1}]] (* Michael Somos, Jul 22 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = prod( k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n))^12; polcoeff( A + 64 * x /A, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A + 64 * x / A, n))} /* Michael Somos, Nov 11 2006 */
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CROSSREFS
| Cf. A007249, A022577.
Sequence in context: A100413 A160344 A163691 * A083936 A160288 A133238
Adjacent sequences: A007244 A007245 A007246 * A007248 A007249 A007250
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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