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%I M5176 #54 Dec 17 2021 11:05:43
%S 1,24,4372,96256,1240002,10698752,74428120,431529984,2206741887,
%T 10117578752,42616961892,166564106240,611800208702,2125795885056,
%U 7040425608760,22327393665024,68134255043715,200740384538624
%N McKay-Thompson series of class 2A for the Monster group with a(0) = 24.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.
%D S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 26.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A007241/b007241.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from T. D. Noe)
%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H Yang-Hui He, John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.
%H J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F G.f. 48 + 64(g_n^(24) + g_n^(-24)) where q = e^(-Pi sqrt(n)) and g_n is Ramanujan's class invariant. - _Michael Somos_, Apr 20 2005
%F a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017
%e G.f. = 1/q + 24 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...
%t a[n0_] := Module[{n=n0, A}, If[n < -1, 0, n++; A = Product[ 1 - x^(2*k-1) , {k, 1, Quotient[n+1, 2]}]^24; SeriesCoefficient[ A + x*48 + x^2*4096/A, {x, 0, n}]]]; Table[ a[n], {n, -1, 16}] (* _Jean-François Alcover_, Oct 16 2012, after _Michael Somos_ *)
%t a[ n_] := SeriesCoefficient[ With[{A = q QPochhammer[ -q, q]^24}, -80 + (1 + 64 A)^2 / A], {q, 0, n}]; (* _Michael Somos_, Apr 06 2015 *)
%t nmax = 50; CoefficientList[Series[48*x + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}] + Product[1/(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 01 2017 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = prod( k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n))^24; polcoeff( A + x*48 + x^2 * 4096/A, n))}; /* _Michael Somos_, Feb 07 2003 */
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^2 + A) / eta(x + A))^24; polcoeff( -80*x + (1 + 64 * x * A)^2 / A, n))}; /* _Michael Somos_, Apr 06 2015 */
%Y A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.
%K nonn,nice
%O -1,2
%A _N. J. A. Sloane_
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