|
%I M5369 #54 Mar 12 2021 22:24:41
%S 1,104,4372,96256,1240002,10698752,74428120,431529984,2206741887,
%T 10117578752,42616961892,166564106240,611800208702,2125795885056,
%U 7040425608760,22327393665024,68134255043715,200740384538624
%N Expansion of 16 * (1 + k^2)^4 /(k * k'^2)^2 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
%C McKay-Thompson series of class 2A for the Monster group with a(0) = 104.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.
%D R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 517.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A007267/b007267.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 from T. D. Noe)
%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 R. Fricke, <a href="http://www.archive.org/details/elliptischenfun02ricrich">Die elliptischen Funktionen und ihre Anwendungen, Vol. 2</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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 Titus Piezas III, <a href="http://www.oocities.org/titus_piezas/Ramanujan_a.pdf">Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins</a>.
%H Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of 16 * (1 + k'^2)^4 /(k' * k^2)^2 in powers of q^2. - _Michael Somos_, Nov 11 2006
%F a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017
%e G.f. = 1/q + 104 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...
%t a[ n_] := If[ n < -1, 0, With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* _Michael Somos_, Jun 29 2011 *)
%t a[ n_] := If[ n < -1, 0, With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* _Michael Somos_, Jun 30 2011 *)
%t QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = (A + 64*(q/A))^2 + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%t nmax = 20; CoefficientList[Series[128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 03 2018 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, A = prod(k=1, n\2 + 1, 1 - x^(2*k - 1),1 + x^2 * O(x^n))^12; polcoeff( (64 * x / A + A)^2, n+1))};
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( (A + 64 * x / A)^2, n))}; /* _Michael Somos_, Nov 11 2006 */
%Y Cf. A007241, A045478. Convolution square of A007247.
%Y A045478, A007241, A106207, A007267, and A101558 are all essentially the same sequence.
%K nonn,nice
%O -1,2
%A _N. J. A. Sloane_, Apr 28 1994
|
