%I #28 Jun 22 2018 15:11:32

%S 1,0,4,2,6,10,15,18,37,30,57,70,105,114,178,192,285,346,465,522,751,

%T 830,1125,1328,1708,1974,2600,2964,3795,4424,5541,6390,8090,9230,

%U 11424,13308,16225,18714,22941,26216,31794,36730,44020,50544,60671,69360,82560,94952

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

%H G. C. Greubel, <a href="/A058613/b058613.txt">Table of n, a(n) for n = -1..2500</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f. T30B = 3 + e30A + 1 / e30A = 1 + e30C + 4 / e30C = -2 + e30D + 1 / e30D = -1 + e30F + 1 / e30F where e30A is g.f. A205826, e30C is g.f. A132321, e30D is g.f. A205962, and e30F is g.f. A205977.

%F Convolution square of A058732. - _Michael Somos_, Feb 02 2012

%F a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%F Expansion of A + 3 + 1/A, where A := (eta(q)*eta(q^6)*eta(q^10)*eta(q^15] )/( eta(q^2)*eta(q^3)*eta(q^5)*eta(q^30)))^3, in powers of q. - _G. C. Greubel_, Jun 22 2018

%e T30B = 1/q + 4*q + 2*q^2 + 6*q^3 + 10*q^4 + 15*q^5 + 18*q^6 + 37*q^7 + ...

%t nmax = 50; QP = QPochhammer; A = x*O[x]^(nmax + 1); A = (QP[A + x^3]*QP[A + x^5]*QP[A + x^6]*QP[A + x^10])/(QP[A + x]*QP[A + x^2]*QP[A + x^15]*QP[A + x^30]); a[n_] := SeriesCoefficient[x^2/A + A - x, n + 1]; Table[a[n], {n, -1, nmax}] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; A := (eta[q]*eta[q^6]*eta[q^10]* eta[q^15]/(eta[q^2]*eta[q^3]*eta[q^5]*eta[q^30]))^3; a:=CoefficientList[ Series[q*(A + 3 + 1/A), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 22 2018 *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = 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)); polcoeff( -x + A + x^2 / A,n))} /* _Michael Somos_, Feb 02 2012 */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, A058732, A132321, A205826, A205962, A205977.

%K nonn

%O -1,3

%A _N. J. A. Sloane_, Nov 27 2000