A058487 McKay-Thompson series of class 12I for the Monster group.

1, 2, 1, 0, -2, -2, 2, 4, 3, -4, -8, -4, 5, 14, 7, -8, -20, -12, 14, 28, 17, -20, -44, -24, 28, 66, 36, -40, -90, -52, 56, 124, 71, -80, -176, -96, 109, 244, 133, -144, -326, -182, 198, 432, 240, -268, -580, -316, 349, 772, 420, -456, -1004, -552, 600, 1300, 713, -780, -1692, -916, 1001, 2186, 1182

Offset

0,2

Comments

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

Number 5 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(12). [Yang 2004] - Michael Somos, Jul 21 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..502 from G. A. Edgar)

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.

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 sequences related to groups

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of (psi(x) / psi(x^3))^2 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 21 2014

G.f.: ( Product_{k>0} (1 - x^(6*k - 2)) * (1 - x^(6*k - 4)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k - 5))) )^2.

Expansion of q^(1/2) * (eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^4)) in powers of q.

Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, 0,...]. - Michael Somos, Mar 18 2004

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + 3*v - u^2*v + v^2. - Michael Somos, Mar 18 2004

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A186924.

a(n) = (-1)^n * A062243(n).

Convolution square is A128633. Convolution inverse of A217786. - Michael Somos, Jul 21 2014

Example

G.f. = 1 + 2*x + x^2 - 2*x^4 - 2*x^5 + 2*x^6 + 4*x^7 + 3*x^8 - 4*x^9 - 8*x^10 + ...
T12I = 1/q + 2*q + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / EllipticTheta[ 2, 0, q^3]^2, {q, 0, 2 n - 1}]; (* Michael Somos, Jul 21 2014 *)
QP = QPochhammer; s = QP[q^2]^4*(QP[q^3]^2/(QP[q]^2*QP[q^6]^4)) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

Prog

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

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, A062243, A128633, A186924, A217786.

Sequence in context: A047654 A358477 A351981 * A062243 A128095 A316658

Adjacent sequences: A058484 A058485 A058486 * A058488 A058489 A058490

Keyword

sign,changed

Author

N. J. A. Sloane, Nov 27 2000

Status

approved