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A035099
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McKay-Thompson series of class 2B for the Monster group with a(0) = 40.
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7
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1, 40, 276, -2048, 11202, -49152, 184024, -614400, 1881471, -5373952, 14478180, -37122048, 91231550, -216072192, 495248952, -1102430208, 2390434947, -5061476352, 10487167336, -21301241856, 42481784514, -83300614144
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OFFSET
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-1,2
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COMMENTS
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Also Fourier coefficients of j_2 where j_2 is an analytic isomorphism H/\Gamma_0(2) ->\hat{C}.
"The function j_2 is analogous to j because it is modular (weight zero) for \Gamma_0(2), holomorphic on the upper half-plane, has a simple pole at infinity, generates the field of \Gamma_0(2)-modular functions, and defines a bijection of a \Gamma_0(2) fundamental set with C." from the Brent article page 260 using his notation of j_2. - Michael Somos, Mar 08 2011
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REFERENCES
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G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
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LINKS
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R. E. Borcherds, Introduction to the monster Lie algebra, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).
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FORMULA
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Expansion of 64 + q^(-1) * (phi(-q) / psi(q))^8 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 08 2011
Expansion of 64 + (eta(q) / eta(q^2))^24 in powers of q. - Michael Somos, Mar 08 2011
j_2 = E_{gamma, 2}^2 / E_{oo, 4} in the notation of Brent where E_{gamma, 2} is g.f. for A004011 and E_{oo, 4} is g.f. for A007331. - Michael Somos, Mar 08 2011
G.f.: 64 + x^(-1) * (Product_{k>0} 1 + x^k)^(-24). - Michael Somos, Mar 08 2011
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Nov 16 2016
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EXAMPLE
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j_2 = 1/q + 40 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
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MATHEMATICA
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max = 21; f[x_] := Product[ 1 + x^k, {k, 1, max}]^(-24); coes = CoefficientList[ Series[ f[x], {x, 0, max} ], x]; a[n_] := coes[[n+2]]; a[0] = 40; Table[a[n], {n, -1, max-1}] (* Jean-François Alcover, Nov 03 2011, after Michael Somos *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 64 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* Michael Somos, Mar 08 2011 */
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CROSSREFS
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KEYWORD
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easy,sign,nice,core
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AUTHOR
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Barry Brent (barryb(AT)primenet.com)
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STATUS
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approved
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