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A007260
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McKay-Thompson series of class 6a for Monster.
(Formerly M5238)
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3
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1, -33, -153, -713, -2550, -7479, -20314, -51951, -122229, -276656, -601068, -1254105, -2541531, -5018721, -9647991, -18168984, -33554784, -60818040, -108471674, -190607871, -330140403, -564580142, -953980392, -1593599832, -2634301308, -4311874755, -6991318008
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OFFSET
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0,2
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COMMENTS
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A more correct name would be: Expansion of replicable function of class 6a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
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FORMULA
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Expansion of q * ((eta(q^2) / eta(q^6))^6 - 27 * (eta(q^6) / eta(q^2))^6) in powers of q^2. - Michael Somos, Jun 14 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -1 / f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 14 2015
a(n) ~ -exp(2*Pi*sqrt(2*n/3)) / (2^(3/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
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EXAMPLE
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G.f. = 1 - 33*x - 153*x^2 - 713*x^3 - 2550*x^4 - 7479*x^5 - 20314*x^6 + ...
T6a = 1/q - 33*q - 153*q^3 - 713*q^5 - 2550*q^7 - 7479*q^9 - 20314*q^11 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^3])^6 - 27 x (QPochhammer[ x^3] / QPochhammer[ x])^6, {x, 0, n} ]; (* Michael Somos, Jun 14 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^6 - 27 * x * (eta(x^3 + A) / eta(x + A))^6, n))}; /* Michael Somos, Jun 14 2015 */
(PARI) { my(q='q+O('q^66), t=(eta(q)/eta(q^3))^6 ); Vec( t - 27*q/t ) } \\ Joerg Arndt, Apr 02 2017
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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