A058543 McKay-Thompson series of class 18e for the Monster group.

1, -2, 1, -4, 8, -6, 10, -16, 18, -26, 33, -40, 58, -74, 82, -112, 147, -166, 212, -268, 316, -392, 476, -560, 695, -838, 967, -1184, 1430, -1648, 1970, -2352, 2731, -3236, 3803, -4404, 5206, -6080, 6984, -8192, 9553, -10942, 12709, -14736, 16886, -19506, 22448, -25648, 29552

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

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

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

Expansion of chi(-x)^2 * chi(-x^3)^2 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Aug 18 2007

Expansion of q^(-1/3) * (eta(q) * eta(q^3) / (eta(q^2) * eta(q^6)))^2 in powers of q. - Michael Somos, Aug 18 2007

Euler transform of period 6 sequence [ -2, 0, -4, 0, -2, 0, ...]. - Michael Somos, Aug 18 2007

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - u^2 * v - 4 * u. - Michael Somos, Aug 18 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 4 / f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 18 2007

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 10*x^6 - 16*x^7 + 18*x^8 - ...
T18e = 1/q - 2*q^2 + q^5 - 4*q^8 + 8*q^11 - 6*q^14 + 10*q^17 - 16*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^3, q^6])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Aug 18 2007 */

Crossrefs

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

Sequence in context: A257706 A118272 A112173 * A353661 A156817 A008301

Adjacent sequences: A058540 A058541 A058542 * A058544 A058545 A058546

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved