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%I #31 Jan 24 2018 09:08:48
%S 1,4,276,2048,11202,49152,184024,614400,1881471,5373952,14478180,
%T 37122048,91231550,216072192,495248952,1102430208,2390434947,
%U 5061476352,10487167336,21301241856,42481784514,83300614144,160791890304
%N McKay-Thompson series of class 4A for the Monster group with a(0) = 4.
%H G. C. Greubel, <a href="/A134786/b134786.txt">Table of n, a(n) for n = -1..1000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Associated with permutations in Mathieu group M24 of shape (4)^4(2)^2(1)^4.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A107080(n) unless n=0. Convolution with A030212 is A037219.
%F a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e G.f. = 1/q + 4 + 276*q + 2048*q^2 + 11202*q^3 + 49152*q^4 + 184024*q^5 + ...
%t a[0] = 4; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 21}] (* _Jean-François Alcover_, Oct 14 2013, after _Michael Somos_ *)
%t QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 20*q + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, after _Michael Somos_ *)
%t a[ n_] := SeriesCoefficient[ -20 + QPochhammer[ -q, q^2]^24 / q, {q, 0, n}]; (* _Michael Somos_, May 05 2016 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) / eta(x^4 + A))^8 / x; polcoeff( 12 + A + 256 / A, n))};
%Y Cf. A030212, A037219, A107080.
%Y Cf. A097340. [From _R. J. Mathar_, Dec 13 2008]
%Y A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
%K nonn
%O -1,2
%A _Michael Somos_, Nov 22 2007
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