A205977 McKay-Thompson series of class 30F for the Monster group with a(0) = 1.

1, 1, 3, 3, 8, 8, 16, 17, 33, 35, 59, 65, 105, 116, 175, 198, 292, 330, 466, 533, 736, 842, 1132, 1304, 1725, 1985, 2576, 2974, 3809, 4394, 5555, 6415, 8030, 9261, 11475, 13234, 16264, 18734, 22843, 26296, 31849, 36613, 44058, 50602, 60551, 69452, 82669

Offset

-1,3

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Formula

Expansion of eta(q^3) * eta(q^5) * eta(q^6) * eta(q^10) / (eta(q) * eta(q^2) * eta(q^15) * eta(q^30)) in powers of q.

Euler transform of period 30 sequence [ 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 2, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058617(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

1/q + 1 + 3*q + 3*q^2 + 8*q^3 + 8*q^4 + 16*q^5 + 17*q^6 + 33*q^7 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(3*k)) * (1-x^(5*k)) * (1-x^(6*k)) * (1-x^(10*k)) / ((1-x^k) * (1-x^(2*k)) * (1-x^(15*k)) * (1-x^(30*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= eta[q^3]*eta[q^5]*eta[q^6]* eta[q^10]/(eta[q]*eta[q^2]*eta[q^15]*eta[q^30]); a:= CoefficientList[ Series[q*A, {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^5 + A) * eta(x^6 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^2 + A) * eta(x^15 + A) * eta(x^30 + A)), n))}

Crossrefs

Cf. A058617.

Sequence in context: A168283 A135291 A058617 * A363725 A238623 A138135

Adjacent sequences: A205974 A205975 A205976 * A205978 A205979 A205980

Keyword

nonn

Author

Michael Somos, Feb 02 2012

Status

approved