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A112142
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McKay-Thompson series of class 8B for the Monster group.
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2
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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, 201825420, 303257512
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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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
| Expansion of chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/2) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^12 in powers of q.
G.f.: Product_{k>0} (1 + (-x)^k)^-12 = Product_{k>0} (1 + x^(2*k - 1))^-12.
a(n) = (-1)^n * A007249(n). Convolution inverse of A124863.
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EXAMPLE
| 1 + 12*x + 66*x^2 + 232*x^3 + 639*x^4 + 1596*x^5 + 3774*x^6 + 8328*x^7 + ...
T8B = 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_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}]^-12, {x, 0, 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^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^12, n))}
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CROSSREFS
| Cf. A007249, A124863.
Sequence in context: A045853 A014787 A007249 * A114243 A000972 A180392
Adjacent sequences: A112139 A112140 A112141 * A112143 A112144 A112145
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Aug 28 2005
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