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A007248 McKay-Thompson series of class 4C for the Monster group.
(Formerly M5084)
6
1, 20, -62, 216, -641, 1636, -3778, 8248, -17277, 34664, -66878, 125312, -229252, 409676, -716420, 1230328, -2079227, 3460416, -5677816, 9198424, -14729608, 23328520, -36567242, 56774712, -87369461, 133321908, -201825396, 303248408, -452431503 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

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

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

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.

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

LINKS

Table of n, a(n) for n=0..28.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

16*(theta_3/theta_2)^4 - 8 = 16 / lambda(z) - 8.

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

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

Expansion of q * (8 + (eta(q) / eta(q^4))^8) in powers of q^2. - Michael Somos, Nov 14 2006

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

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

a(n) = A029845(2*n - 1) = A124972(2*n - 1). - Michael Somos, Nov 14 2006.

EXAMPLE

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

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

MATHEMATICA

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

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

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

PROG

(PARI) 8*x + prod(n=1, 39, if( n%4, 1 - x^n, 1), 1 + O(x^40))^8

(PARI) {a(n) = local(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 */

CROSSREFS

Cf. A029845, A097243, A124972.

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

Sequence in context: A041784 A105092 A112144 * A117431 A159504 A117432

Adjacent sequences:  A007245 A007246 A007247 * A007249 A007250 A007251

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 18 06:09 EST 2014. Contains 252079 sequences.