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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,2
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COMMENTS
| 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), holomorphics 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
Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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).
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
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LINKS
| T. D. Noe, Table of n, a(n) for n=-1..1000
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.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
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
| 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_{\infty, 4} in the notation of Brent where E_{\gamma, 2} is g.f. for A004011 and E_{\infty, 4} is g.f. for A007331. - Michael Somos Mar 08 2011
G.f.: 64 + q^(-1) * (Product_{k>0} 1 + x^k)^(-24). - Michael Somos Mar 08 2011
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EXAMPLE
| j_2 = 1/q + 40 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
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MATHEMATICA
| 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}] (* From Jean-François Alcover, Nov 03 2011, after Michael Somos *)
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PROG
| (PARI) {a(n) = local(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
| A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
Sequence in context: A073962 A115170 A117216 * A065255 A061993 A087954
Adjacent sequences: A035096 A035097 A035098 * A035100 A035101 A035102
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KEYWORD
| easy,sign,nice
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AUTHOR
| Barry Brent (barryb(AT)primenet.com)
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