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A045488 McKay-Thompson series of class 6E for the Monster group with a(0) = 1. 4
1, 1, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744, -20448, -46944, -26916 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

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, Comm. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Morris Newman, Construction and application of a class of modular functions. II. Proc. London Math. Soc. (3) 9 1959 373-387.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of (1/q) * a(q^2) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 22 2015

Expansion of 6 + eta(q)^5 * eta(q^3) / (eta(q^2) * eta(q^6)^5) in powers of q. - Michael Somos, May 22 2015

a(n) = A007258(n) = A105559(n) = A128632(n) = A128633(n) = A258094(n) unless n=0. - Michael Somos, May 22 2015

EXAMPLE

G.f. = 1/q + 1 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 6 + QPochhammer[ q]^5 QPochhammer[ q^3] / (q QPochhammer[ q^2] QPochhammer[ q^6]^5), {q, 0, n}]; (* Michael Somos, May 22 2015 *)

a[ n_] := SeriesCoefficient[ -2 + (1/q) (QPochhammer[ q^2] QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^6]^3))^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3] + 3 EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q]) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 6*x + eta(x + A)^5 * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)^5), n))}; /* Michael Somos, May 22 2015 */

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -2*x + (eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3))^3, n))}; /* Michael Somos, May 22 2015 */

CROSSREFS

Cf. A007258, A105559, A128632, A128633, A258094.

Sequence in context: A268818 A112148 A007258 * A082530 A099404 A095156

Adjacent sequences:  A045485 A045486 A045487 * A045489 A045490 A045491

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 22 18:06 EST 2019. Contains 319365 sequences. (Running on oeis4.)