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A007248 McKay-Thompson series of class 4C for the Monster group.
(Formerly M5084)
9

%I M5084

%S 1,20,-62,216,-641,1636,-3778,8248,-17277,34664,-66878,125312,-229252,

%T 409676,-716420,1230328,-2079227,3460416,-5677816,9198424,-14729608,

%U 23328520,-36567242,56774712,-87369461,133321908,-201825396,303248408,-452431503

%N McKay-Thompson series of class 4C for the Monster group.

%D J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%D D. Ford, J. McKay and S. P. Norton, ``More on replicable functions,'' Commun. Algebra 22, No. 13, 5175-5193 (1994).

%D J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

%D McKay, John; Strauss, Hubertus. The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A007248/b007248.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: 16*(theta_3/theta_2)^4 - 8 = 16 / lambda(z) - 8.

%F G.f.: 8*x^(1/2) + (Product_{k>0} (1 - x^(k/2)) / (1 - x^(2*k)))^8.

%F Expansion of q * ( -8 + 16 / lambda(z)) in powers of q^2 where nome q = exp(Pi i z). - _Michael Somos_, Nov 14 2006

%F Expansion of 4 * q^(1/2) * (k(q) + 1 / k(q)) in powers of q where nome q = exp(Pi i z). - _Michael Somos_, Nov 11 2014

%F Expansion of q * (8 + (eta(q) / eta(q^4))^8) in powers of q^2. - _Michael Somos_, Nov 14 2006

%F Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v + 24)^2 - (v + 8) * u^2. - _Michael Somos_, Nov 14 2006

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A097243. - _Michael Somos_, Jul 22 2011

%F a(n) = A029845(2*n - 1) = A124972(2*n - 1). - _Michael Somos_, Nov 14 2006.

%e G.f. = 1 + 20*x - 62*x^2 + 216*x^3 - 641*x^4 + 1636*x^5 - 3778*x^6 + ...

%e T4C = 1/q + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...

%t a[ n_] := SeriesCoefficient[ 8 q + (QPochhammer[ q] / QPochhammer[ q^4])^8, {q, 0, 2 n}]; (* _Michael Somos_, Jul 22 2011 *)

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ -8 + 16 / m, {q, 0, 2 n - 1}]]; (* _Michael Somos_, Jul 22 2011 *)

%t a[ n_] := SeriesCoefficient[ -8 + 16 (EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q])^4, {q, 0, 2 n - 1}]; (* _Michael Somos_, Jul 22 2011 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8*x + (eta(x + A) / eta(x^4 + A))^8, n))}; /* _Michael Somos_, Nov 14 2006 */

%Y Cf. A029845, A097243, A124972.

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified December 11 10:48 EST 2018. Contains 318049 sequences. (Running on oeis4.)