Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #41 Aug 03 2020 00:46:03
%S 1,-8,276,-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 with a(0) = -8.
%C Unsigned sequence gives McKay-Thompson series of class 4A for Monster; also character of extremal vertex operator algebra of rank 12.
%C The value of a(0) is the Rademacher constant for the modular function and appears in Conway and Norton's Table 4. - _Michael Somos_, Mar 08 2011
%D G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
%H T. D. Noe, <a href="/A045479/b045479.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="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of 16 + (eta(q) / eta(q^2))^24 in powers of q. - _Michael Somos_, Mar 08 2011
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e 1/q - 8 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
%t a[0] = -8; 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 QP = QPochhammer; s = 16*q + (QP[q]/QP[q^2])^24 + 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( 16 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* _Michael Somos_, Mar 08 2011 */
%Y A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
%K sign,easy,nice
%O -1,2
%A _N. J. A. Sloane_