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%I M5353 #67 Feb 14 2021 01:16:32
%S 1,-76,-702,-5224,-23425,-98172,-336450,-1094152,-3188349,-8913752,
%T -23247294,-58610304,-140786308,-328793172,-740736900,-1629664840,
%U -3486187003,-7307990208,-14976155896,-30157221352,-59594117256,-115975615160,-222119374922,-419704427016
%N McKay-Thompson series of class 4a for the Monster group.
%C A more correct name would be: Expansion of replicable function of class 4a. See Alexander et al., 1992. - _N. J. A. Sloane_, Jun 12 2015
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A007250/b007250.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..499 from G. A. Edgar)
%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="/A007242/a007242_1.pdf">Completely replicable functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">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 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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = - f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 22 2011
%F a(n) = A007249(n) - 64 * A022577(n-1).
%F Expansion of q^(1/2) * ((eta(q) / eta(q^2))^12 - 64*(eta(q^2) / eta(q))^12) in powers of q. - _G. A. Edgar_, Mar 10 2017
%e G.f. = 1 - 76*x - 702*x^2 - 5224*x^3 - 23425*x^4 - 98172*x^5 - 336450*x^6 + ...
%e T4a = 1/q - 76*q - 702*q^3 - 5224*q^5 - 23425*q^7 - 98172*q^9 - ...
%p A022577L := proc(n)
%p mul((1+x^m)^12,m=1..n+1) ;
%p taylor(%,x=0,n+1) ;
%p gfun[seriestolist](%) ;
%p end proc:
%p A007249L := proc(n)
%p if n = 0 then
%p 0 ;
%p else
%p mul(1/(1+x^m)^12,m=1..n+1) ;
%p taylor(%,x=0,n+1) ;
%p gfun[seriestolist](%) ;
%p end if;
%p end proc:
%p a022577 := A022577L(80) ;
%p a007249 := A007249L(80) ;
%p printf("1,");
%p for i from 1 to 78 do
%p printf("%d,", op(i+1,a007249)-64*op(i,a022577) );
%p end do: # _R. J. Mathar_, Sep 30 2011
%t a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e - 64 / e) q^(1/2), {q, 0, n}]]; (* _Michael Somos_, Jul 22 2011 *)
%t QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = A - 64*(q/A) + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)
%t nmax = 30; CoefficientList[Series[Product[((1-x^k) / (1-x^(2*k)))^12, {k, 1, nmax}] - 64*x*Product[((1-x^(2*k)) / (1-x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 11 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A - 64 * x / A, n))}; /* _Michael Somos_, Jul 22 2011 */
%o (PARI) N=66; q='q+O('q^N); t=(eta(q)/eta(q^2))^12; Vec(t - 64*q/t) \\ _Joerg Arndt_, Mar 11 2017
%Y Cf. A007242, A007249, A007260, A022577.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
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