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%I M5434 #47 Mar 12 2021 22:24:41
%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
%N McKay-Thompson series of class 2B for the Monster group.
%C Unsigned sequence gives McKay-Thompson series of class 4A for the Monster group; also character of extremal vertex operator algebra of rank 12.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see pp. 139, 424.
%D G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007246/b007246.txt">Table of n, a(n) for n = -1..1000</a>
%H R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/">Introduction to the monster Lie algebra</a>, pp. 99-107 of M. Liebeck and J. Saxl, editors, Groups, Combinatorics and Geometry (Durham, 1990). London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, 1992.
%H B. Brent, <a href="https://projecteuclid.org/euclid.em/1047674207">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3, 257-274.
%H J. H. Conway and S. P. Norton, <a href="https://doi.org/10.1112/blms/11.3.308">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 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 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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%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 Expansion of 24 + chi(-q)^24 / q in powers of q where chi() is a Ramanujan theta function.
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e T2B = 1/q + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 - ...
%t a[0] = 0; a[n_] := SeriesCoefficient[ Product[1 - q^k, {k, 1, n+1, 2}]^24/q, {q, 0, n}]; Table[a[n], {n, -1, 20}] (* _Jean-François Alcover_, Oct 14 2013, after _Michael Somos_ *)
%t a[ n_] := SeriesCoefficient[ 24 + 1/q QPochhammer[ q, q^2]^24, {q, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 24 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* _Michael Somos_, Jul 05 2014 */
%Y A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
%K sign,easy,nice
%O -1,3
%A _N. J. A. Sloane_
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