A058537 McKay-Thompson series of class 18b for the Monster group.

1, 7, 8, 22, 42, 63, 106, 190, 267, 428, 652, 932, 1367, 2017, 2774, 3950, 5539, 7541, 10342, 14184, 18889, 25435, 33974, 44720, 58952, 77550, 100546, 130780, 169273, 217230, 278636, 356566, 452544, 574548, 726938, 914742, 1149685, 1441787, 1798740, 2242436

Offset

0,2

Comments

Convolution inverse is A258941. - Vaclav Kotesovec, Nov 07 2015

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

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

Michael Somos, Emails to N. J. A. Sloane, 1993

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/6) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012

Expansion of q^(1/6) * a(q) / (b(q) * c(q)/3)^(1/2) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Aug 20 2012

Convolution square is A058092. Convolution sixth power is A030197. - Michael Somos, Jun 16 2012

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015

Example

1 + 7*x + 8*x^2 + 22*x^3 + 42*x^4 + 63*x^5 + 106*x^6 + 190*x^7 + 267*x^8 + ...
T18b = 1/q + 7*q^5 + 8*q^11 + 22*q^17 + 42*q^23 + 63*q^29 + 106*q^35 + ...

Mathematica

CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3) / (QPochhammer[x, x]*QPochhammer[x^3, x^3]^2), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(-1/6)*eta[q]*eta[q^3]^2/(eta[q]^3 + 9*eta[q^9]^3); CoefficientList[Series[1/A, {q, 0, 60}], q] (* G. C. Greubel, Jun 22 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/6), n))} \\ Michael Somos, Jun 16 2012
(PARI) q='q+O('q^50); A = (eta(q)^3 + 9*q*eta(q^9)^3)/(eta(q)* eta(q^3)^2); Vec(A) \\ G. C. Greubel, Jun 22 2018

Crossrefs

Cf. A030197, A058092, A258941.

Sequence in context: A322651 A325322 A288068 * A002362 A042417 A042845

Adjacent sequences: A058534 A058535 A058536 * A058538 A058539 A058540

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved