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%I #26 Feb 05 2020 08:53:48
%S 1,0,0,4371,96256,1143745,9646891,64680601,366845011,1829005611,
%T 8223700027,33950840617,130416170627,470887671187,1610882889457,
%U 5254605009307,16428803075153,49446546607298,143782211788218
%N q-expansion of character of vertex-operator superalgebra of rank 23.5 on which Baby Monster group acts.
%D T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 423.
%D G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
%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>).
%F Let X=sqrt( Sum (q^(m^2/2), m=-inf..inf) / q^(1/24) Product( 1-q^n, n=1..inf ) ). Then series is X^47 - 47*X^23.
%F a(n) ~ 47^(1/4) * exp(Pi*sqrt(47*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Feb 05 2020
%e G.f. = 1 + 4371*x^3 + 96256*x^4 + 1143745*x^5 + 9646891*x^6 + 64680601*x^7 + ...
%e G.f. = q^(-47/48) * (1 + 4371 * q^(3/2) + 96256 * q^2 + 1143745 * q^(5/2) + ...).
%t nmax = 30; CoefficientList[Series[Product[(1 + x^(2*k + 1))^47, {k, 0, nmax}] - 47*x*Product[(1 + x^(2*k + 1))^23, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 05 2020 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = prod( i=1, (1+n)\2, 1 + x^(2*i-1), 1 + x * O(x^n)); polcoeff( A^47 - 47 * x * A^23, n))}; /* _Michael Somos_, Jul 01 2004 */
%Y Cf. A007245 (rank 8), A097340 (rank 12), A028523 (rank 14), A028524 (rank 15), A028525 (rank 15.5), A000521 (rank 24).
%K nonn
%O 0,4
%A _N. J. A. Sloane_
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