%I #37 Oct 04 2019 11:57:03
%S 1,0,276,2048,11202,49152,184024,614400,1881471,5373952,14478180,
%T 37122048,91231550,216072192,495248952,1102430208,2390434947,
%U 5061476352,10487167336,21301241856,42481784514,83300614144,160791890304,305854488576,573872089212,1063005978624,1945403602764,3519965179904
%N McKay-Thompson series of class 4A for the Monster group.
%C Also character of extremal vertex operator algebra of rank 12.
%H Seiichi Manyama, <a href="/A107080/b107080.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from T. D. Noe)
%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 G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (<a href="http://www.math.ksu.edu/~gerald/papers/dr.pdf">pdf</a>, <a href="http://www.math.ksu.edu/~gerald/papers/dr.ps.gz">ps</a>).
%H G. Hoehn, <a href="https://arxiv.org/abs/0706.0236">Selbstduale Vertexoperatorsuperalgebren und das Babymonster</a>, arXiv:0706.0236 [math.QA], 2007, from Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F G.f.: (1/x)(Product_{k>0} (1+x^k)/(1+x^(2k)))^24 -24.
%F a(n) = -(-1)^n * A007246(n).
%F a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015
%e T4A = 1/q + 276q + 2048q^2 + 11202q^3 + 49152q^4 + 184024q^5 +...
%t a[0] = 0; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}] // Abs; Table[a[n], {n, -1, 20}] (* _Jean-François Alcover_, Oct 14 2013, after _Michael Somos_ *)
%t QP = QPochhammer; s = (QP[q^2]^2/QP[q]/QP[q^4])^24 - 24*q + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, after _Michael Somos_ *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^24 - 24*x, n))};
%o (PARI) q='q+O('q^66); Vec(+40*q+(eta(q)^4 / eta(q^4)^4 - q*4^2*eta(q^4)^4 / eta(q)^4)^2) \\ _Joerg Arndt_, Mar 23 2017
%Y Cf. A007246.
%Y A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
%K nonn
%O -1,3
%A _Michael Somos_, May 11 2005