A058658 McKay-Thompson series of class 38a for Monster.

1, 2, 1, 3, 4, 7, 8, 13, 15, 21, 26, 35, 42, 56, 67, 86, 106, 132, 158, 199, 236, 290, 346, 420, 500, 603, 711, 850, 1002, 1189, 1393, 1648, 1922, 2258, 2629, 3075, 3566, 4153, 4801, 5569, 6421, 7420, 8528, 9831, 11268, 12942, 14801, 16949, 19337, 22090, 25140, 28644, 32536, 36978

Offset

-1,2

Links

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

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of (T19A + 4)^(1/2), where T19A = A058549, in powers of q. - G. C. Greubel, Jun 24 2018

a(n) ~ exp(2*Pi*sqrt(2*n/19)) / (2^(3/4) * 19^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 27 2018

Example

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 + ...

Mathematica

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A075616 A160186 A050041 * A070827 A160191 A268613

Adjacent sequences: A058655 A058656 A058657 * A058659 A058660 A058661

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

Terms a(12) onward added by G. C. Greubel, Jun 24 2018

Status

approved