A062243 McKay-Thompson series of class 24c 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

Expansion of a Hauptmodul for Gamma'_0(12).

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Euler transform of period 12 sequence [-2, 0, 0, -2, -2, 0, -2, -2, 0, 0, -2, 0, ...]. - Michael Somos, May 14 2004

G.f.: ( Product_{k>0} (1 - x^(4*k)) * (1 - x^(2*k-1)) / (1 - x^(3*k)) )^2.

Given G.f. A(x), then B(q) = A(q^2)^2 / (3*q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u + v)^2 * (u^2 + v^2 - u*v) + 3 * (u^3 + v^3) * (1 + u*v) - 9*u*v * (1 + (u*v)^2) - 90*(u*v)^2 - 27*u*v * (u + v) * (1 + u*v). - Michael Somos, May 14 2004

Expansion of q^(1/2) * (eta(q) * eta(q^4) * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)))^2 in powers of q.

a(n) = (-1)^n * A058487(n).

Power series expansion of f(-x^2)^2 / f(x, x^5)^2 = psi(-x)^2 / psi(-x^3)^2 in powers of x where f(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 22 2017

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 + ...
T24c = 1/q - 2*q + q^3 - 2*q^7 + 2*q^9 + 2*q^11 - 4*q^13 + 3*q^15 + 4*x^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(3/2)]^2, {x, 0, n}]; (* Michael Somos, Oct 22 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^2, n))};

Crossrefs

Cf. A058487.

Sequence in context: A358477 A351981 A058487 * A128095 A316658 A189962

Adjacent sequences: A062240 A062241 A062242 * A062244 A062245 A062246

Keyword

sign

Author

N. J. A. Sloane, Jul 01 2001

Status

approved