A205977 - OEIS

%I #11 Jun 17 2018 22:06:12

%S 1,1,3,3,8,8,16,17,33,35,59,65,105,116,175,198,292,330,466,533,736,

%T 842,1132,1304,1725,1985,2576,2974,3809,4394,5555,6415,8030,9261,

%U 11475,13234,16264,18734,22843,26296,31849,36613,44058,50602,60551,69452,82669

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

%H G. C. Greubel, <a href="/A205977/b205977.txt">Table of n, a(n) for n = -1..1000</a>

%F 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.

%F 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, ...].

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

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

%F a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

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

%t 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 *)

%t 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 *)

%o (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))}

%Y Cf. A058617.

%K nonn

%O -1,3

%A _Michael Somos_, Feb 02 2012