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A007240
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McKay-Thompson series of class 1A for the Monster group with a(0) = 24.
(Formerly M5179)
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205
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1, 24, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184
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OFFSET
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-1,2
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COMMENTS
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Changing the term 24 to 744 gives the classical j-function: see A000521 for much more information.
"The most natural normalization [of the j function] is to set the constant term equal to 24, the number given by Rademacher's infinite series for coefficients of the j function". [Borcherds]
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REFERENCES
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Alexander, D.; Cummins, C.; McKay, J.; and Simons, C.; Completely replicable functions, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165. - N. J. A. Sloane, Jul 22 2012
H. Cohen, Course in Computational Number Theory, page 379.
M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. van Wijngaarden, On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (1953), 389-400 [ gives 100 terms ].
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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.
Hisanori Mishima, Factorizations of many number sequences: primorial - 1, Elliptic modular function j(tau), n=-1 to 100, n=101 to 200, n=201 to 300, n=401 to 500, n=501 to 600, n=601 to 700, n=701 to 800, n=801 to 900, n=901 to 1000.
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FORMULA
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Expansion of j(q) - 720 = Theta_Leech(q) / eta(q)^24 in powers of q. Convolution quotient of A008408 and A007240. - Michael Somos, May 05 2012
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EXAMPLE
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G.f. = 1/q + 24 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
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MATHEMATICA
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Join[{1, 24}, List @@ Expand[ Normal[ Series[ 1728 * KleinInvariantJ[tau], {tau, 0, 29}]]] /. tau -> 1] // Delete[{{3}, {5}}] (* Jean-François Alcover, Sep 27 2015 *)
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PROG
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(PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)) - 720, n))}; /* Michael Somos, May 05 2012 */
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = eta(x + x * O(x^n))^24; polcoeff( (1 + 65520 / 691 * (sum( k=1, n, sigma(k, 11) * x^k) - x * A)) / A, n))}; /* Michael Somos, May 05 2012 */
(PARI) q='q+O('q^66); Vec(ellj(q)-720) \\ Joerg Arndt, Apr 24 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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