A058658 - OEIS

%I #21 Jun 27 2018 11:36:28

%S 1,2,1,3,4,7,8,13,15,21,26,35,42,56,67,86,106,132,158,199,236,290,346,

%T 420,500,603,711,850,1002,1189,1393,1648,1922,2258,2629,3075,3566,

%U 4153,4801,5569,6421,7420,8528,9831,11268,12942,14801,16949,19337,22090,25140,28644,32536,36978

%N McKay-Thompson series of class 38a for Monster.

%H G. C. Greubel, <a href="/A058658/b058658.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>, Comm. 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 (T19A + 4)^(1/2), where T19A = A058549, in powers of q. - _G. C. Greubel_, Jun 24 2018

%F a(n) ~ exp(2*Pi*sqrt(2*n/19)) / (2^(3/4) * 19^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 27 2018

%e T38a = 1/q + 2*q + q^3 + 3*q^5 + 4*q^7 + 7*q^9 + 8*q^11 + 13*q^13 + 15*q^15 + ...

%t QP := QPochhammer; nmax = 100; G[q_] := 1/(QP[q, q^5]*QP[q^4, q^5]); H[q_] := 1/(QP[q^2, q^5]*QP[q^3, q^5]); T19A := -3 + (G[q]*G[q^19] + q^4*H[q]*H[q^19])^3/q; a:= CoefficientList[Series[(q*(T19A + 4) + O[q]^nmax)^(1/2), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 24 2018 *)

%o (PARI) q='q+O('q^40);  A = (eta(q)*eta(q^19)/(eta(q^2)*eta(q^38)))^2; B = - (eta(-q)*eta(-q^19)/(eta(q^2)*eta(q^38)))^2; T19A = (q^4/(A*B) - (A + B)/(4*q))^3; v=Vec((q^2 + T19A)^(1/2)); vector(#v\2, n, v[2*n-1]) \\ _G. C. Greubel_, Jun 25 2018

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

%K nonn

%O -1,2

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

%E Terms a(12) onward added by _G. C. Greubel_, Jun 24 2018