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%I #58 Apr 29 2023 08:09:29
%S 1,40,276,-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) = 40.
%C Also Fourier coefficients of j_2 where j_2 is an analytic isomorphism H/\Gamma_0(2) ->\hat{C}.
%C "The function j_2 is analogous to j because it is modular (weight zero) for \Gamma_0(2), holomorphic on the upper half-plane, has a simple pole at infinity, generates the field of \Gamma_0(2)-modular functions, and defines a bijection of a \Gamma_0(2) fundamental set with C." from the Brent article page 260 using his notation of j_2. - _Michael Somos_, Mar 08 2011
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.
%H T. D. Noe, <a href="/A035099/b035099.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; see also <a href="https://projecteuclid.org/journals/experimental-mathematics/volume-7/issue-3/Quadratic-minima-and-modular-forms/em/1047674207.full">Project Euclid</a>
%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 Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MonsterGroup.html">Monster Group</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F Expansion of 64 + q^(-1) * (phi(-q) / psi(q))^8 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Mar 08 2011
%F Expansion of 64 + (eta(q) / eta(q^2))^24 in powers of q. - _Michael Somos_, Mar 08 2011
%F j_2 = E_{gamma, 2}^2 / E_{oo, 4} in the notation of Brent where E_{gamma, 2} is g.f. for A004011 and E_{oo, 4} is g.f. for A007331. - _Michael Somos_, Mar 08 2011
%F G.f.: 64 + x^(-1) * (Product_{k>0} 1 + x^k)^(-24). - _Michael Somos_, Mar 08 2011
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2016
%e j_2 = 1/q + 40 + 276*q - 2048*q^2 + 11202*q^3 - 49152*q^4 + 184024*q^5 + ...
%t max = 21; f[x_] := Product[ 1 + x^k, {k, 1, max}]^(-24); coes = CoefficientList[ Series[ f[x], {x, 0, max} ], x]; a[n_] := coes[[n+2]]; a[0] = 40; Table[a[n], {n, -1, max-1}] (* _Jean-François Alcover_, Nov 03 2011, after _Michael Somos_ *)
%t QP = QPochhammer; s = 64*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( 64 * x + (eta(x + A) / eta(x^2 + A))^24, n))}; /* _Michael Somos_, Mar 08 2011 */
%Y Cf. A134786, A045479, A007191, A097340, A035099, A007246, A107080 are all essentially the same sequence.
%K easy,sign,nice,core
%O -1,2
%A Barry Brent (barryb(AT)primenet.com)
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