A058729 McKay-Thompson series of class 60E for the Monster group.

1, 1, 1, 1, 1, 3, 3, 4, 4, 4, 7, 8, 10, 11, 12, 16, 18, 22, 25, 28, 34, 38, 45, 51, 58, 69, 77, 88, 99, 112, 131, 146, 165, 184, 206, 238, 266, 298, 331, 368, 418, 465, 520, 576, 637, 714, 791, 880, 973, 1074, 1194, 1316, 1455, 1604, 1766, 1954, 2145, 2360

Offset

0,6

Comments

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

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of chi(-x^3) * chi(-x^15) / (chi(-x) * chi(-x^5)) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jun 09 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (120 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 09 2012

Convolution square is A205962. - Michael Somos, Jun 09 2012

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Expansion of q^(1/2)*eta(q^2)*eta(q^3)*eta(q^10)*eta(q^15)/(eta(q)* eta(q^5)*eta(q^6)*eta(q^30)) in powers of q. - G. C. Greubel, Jun 19 2018

Example

1 + x + x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 7*x^10 + ...
T60E = 1/q + q + q^3 + q^5 + q^7 + 3*q^9 + 3*q^11 + 4*q^13 + 4*q^15 + 4*q^17 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)* eta[q^2]*eta[q^3]*eta[q^10]*eta[q^15]/(eta[q]* eta[q^5]*eta[q^6] *eta[q^30]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 19 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^30 + A)), n))} /* Michael Somos, Jun 09 2012 */

Crossrefs

Cf. A205962.

Sequence in context: A136546 A278765 A210881 * A021303 A303821 A240622

Adjacent sequences: A058726 A058727 A058728 * A058730 A058731 A058732

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved