A058570 - OEIS

%I #30 Jun 28 2018 11:08:50

%S 1,0,4,7,13,19,33,47,74,106,154,214,307,417,575,772,1045,1379,1837,

%T 2394,3135,4048,5232,6686,8560,10840,13737,17273,21701,27086,33783,

%U 41890,51893,63969,78748,96536,118196,144146,175561,213122,258327,312202

%N McKay-Thompson series of class 23A for Monster.

%C Also, McKay-Thompson series of class 23B for Monster. - _Michel Marcus_, Feb 18 2014

%H G. C. Greubel, <a href="/A058570/b058570.txt">Table of n, a(n) for n = -1..1000</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 Expansion of (F + 1)*(F^2 + 4)/F^2, where F = eta(q)*eta(q^23)/(eta(q^2)* eta(q^46)), in powers of q. - _G. C. Greubel_, Jun 14 2018

%F a(n) ~ exp(4*Pi*sqrt(n/23)) / (sqrt(2) * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T23A = 1/q + 4*q + 7*q^2 + 13*q^3 + 19*q^4 + 33*q^5 + 47*q^6 + 74*q^7 + ...

%t nmax = 50; QP = QPochhammer; s = -x + Sum[x^(2*j^2 + j*k + 3*k^2), {j, -nmax, nmax}, {k, -nmax, nmax}]/(QP[x]*QP[x^23]) + O[x]^nmax; CoefficientList[s, x] (* _Jean-François Alcover_, Nov 15 2015, adapted from g.f. in A134781 *)

%t eta[q_] := q^(1/24)*QPochhammer[q]; e46A:= (eta[q]*eta[q^23]/(eta[q^2]* eta[q^46])); a[n_]:= SeriesCoefficient[(e46A + 1)*(4 + e46A^2)/(e46A)^2, {q, 0, n}]; Table[a[n], {n,-1,50}] (* _G. C. Greubel_, Feb 13 2018 *)

%o (PARI) q='q+O('q^50); F = eta(q)*eta(q^23)/(q*eta(q^2)* eta(q^46)); Vec((F+1)*(F^2+4)/F^2) \\ _G. C. Greubel_, Jun 14 2018

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%Y Cf. A134781 (same sequence except for n=0).

%K nonn

%O -1,3

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

%E More terms from _Michel Marcus_, Feb 18 2014