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A007249
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McKay-Thompson series of class 4D for the Monster group.
(Formerly M4846)
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5
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1, -12, 66, -232, 639, -1596, 3774, -8328, 17283, -34520, 66882, -125568, 229244, -409236, 716412, -1231048, 2079237, -3459264, 5677832, -9200232, 14729592, -23325752, 36567222, -56778888, 87369483, -133315692
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The convolution square root of A007191, and also the left and right borders of the triangle A161196. [Gary W. Adamson, Jun 06 2009]
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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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
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: prod(m>=1, 1 + x^m )^(-12).
Expansion of chi(-x)^12 in powers of x where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 64/f(t) where q = exp(2*pi*i*t). - Michael Somos, Jul 22 2011
a(n) = (-1)^n * A112142(n). (class 8B). Convolution inverse of A022577. - Michael Somos, Jul 22 2011
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EXAMPLE
| 1 - 12*x + 66*x^2 - 232*x^3 + 639*x^4 - 1596*x^5 + 3774*x^6 + ...
T4D = 1/q - 12*q + 66*q^3 - 232*q^5 + 639*q^7 - 1596*q^9 + 3774*q^11 - ...
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MATHEMATICA
| a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m / 16 / q)^(1/2), {q, 0, n}]] (* Michael Somos Jul, 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/2) / (m / 16 / q), {q, 0, 2 n}]] (* Michael Somos, Jul 22 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^12, n))} /* Michael Somos, Jul 22 2011 */
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CROSSREFS
| Cf. A007191, A022577, A112142.
Sequence in context: A080559 A045853 A014787 * A112142 A114243 A000972
Adjacent sequences: A007246 A007247 A007248 * A007250 A007251 A007252
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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