A226015 McKay-Thompson series of class 21D for the Monster group with a(0) = 2.

1, 2, 5, 8, 16, 26, 44, 66, 104, 152, 229, 324, 469, 652, 916, 1250, 1716, 2306, 3108, 4116, 5464, 7156, 9373, 12144, 15725, 20190, 25889, 32952, 41881, 52904, 66716, 83688, 104785, 130608, 162486, 201336, 249006, 306874, 377482, 462860, 566513, 691404

Offset

-1,2

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Formula

Expansion of (eta(q^3) * eta(q^7) / (eta(q) * eta(q^21)))^2 in powers of q.

Euler transform of period 21 sequence [ 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 3) - (u+v) * (u^2 + 3 * u*v + v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = f(t) where q = exp(2 Pi i t).

G.f.: 1/x * (Product_{k>0} (1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2.

a(n) = A058566(n) unless n=0.

a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

1/q + 2 + 5*q + 8*q^2 + 16*q^3 + 26*q^4 + 44*q^5 + 66*q^6 + 104*q^7 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[((1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^3] *eta[q^7]/(eta[q]*eta[q^21]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^21 + A)))^2, n))}

Crossrefs

Cf. A058566.

Sequence in context: A129299 A171238 A096541 * A328547 A137685 A169826

Adjacent sequences: A226012 A226013 A226014 * A226016 A226017 A226018

Keyword

nonn

Author

Michael Somos, May 22 2013

Status

approved