A105559 McKay-Thompson series of class 6E for the Monster group with a(0) = 3.

1, 3, 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

Offset

-1,2

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).

Number 2 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(6). [Yang 2004] - Michael Somos, Jul 21 2014

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..147 from G. A. Edgar)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of (eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3))^3 in powers of q.

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = v^2 + 8*u + 6*u*v - u^2*v.

G.f.: x^-1 (Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k - 1)))^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128643.

Expansion of (c(q) / c(q^2))^3 in powers of q where c() is a cubic AGM theta function.

Expansion of q^(-1) * (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 6 sequence [ 3, 0, -6, 0, 3, 0, ...].

a(n) = A007258(n) unless n=0. Convolution inverse of A123633.

Convolution cube of A062242. - Michael Somos, Apr 24 2015

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 3, n, 6}] / Product[ 1 - q^k, {k, 1, n, 2}]^3)^3 / q, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3) )^3, n))};

Crossrefs

Cf. A007258, A062242, A123633.

Sequence in context: A231737 A140072 A187148 * A090038 A308291 A006464

Adjacent sequences: A105556 A105557 A105558 * A105560 A105561 A105562

Keyword

sign

Author

Michael Somos, Apr 13 2005, Jan 21 2009

Status

approved