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A058537 McKay-Thompson series of class 18b for the Monster group. 4
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 (list; graph; refs; listen; history; text; internal format)
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).

M. 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

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Last modified October 18 18:56 EDT 2019. Contains 328197 sequences. (Running on oeis4.)