A007254 McKay-Thompson series of class 6A for Monster. (Formerly M5355)

1, 0, 79, 352, 1431, 4160, 13015, 31968, 81162, 183680, 412857, 864320, 1805030, 3564864, 7000753, 13243392, 24805035, 45168896, 81544240, 143832672, 251550676, 432030080, 735553575, 1233715328, 2052941733

Offset

-1,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..1000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

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

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Wikipedia, Generalized Monstrous Moonshine

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) = A121665(n) + A226235(n) = A121666(n) + 64*A123653(n) = A121667(n) + 81*A284607(n) for n > 0. - Seiichi Manyama, Mar 30 2017

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 30 2017

Example

T6A = 1/q + 79*q + 352*q^2 + 1431*q^3 + 4160*q^4 + 13015*q^5 + 31968*q^6 + ...

Mathematica

nmax = 50; Flatten[{1, 0, Rest[Rest[CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^12, {k, 1, nmax}] + x^2*Product[((1 + x^(3*k))/(1 + x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x]]]}] (* Vaclav Kotesovec, Mar 30 2017 *)
eta[q_] := q^(1/24)*QPochhammer[q]; e6B:= (eta[q^2]*eta[q^3]/(eta[q]* eta[q^6]))^12; a:= CoefficientList[Series[q*(e6B - 12 + 1/e6B), {q, 0, 50}], q]; Table[a[[n]], {n, 1, 50}] (*G. C. Greubel, May 10 2018 *)

Prog

(PARI) q='q+O('q^50); F =(eta(q^2)*eta(q^3)/(eta(q)*eta(q^6)))^12/q;  Vec(F -12 +1/F) \\ G. C. Greubel, May 10 2018

Crossrefs

Cf. A045484.

Sequence in context: A157507 A142897 A142330 * A142439 A200847 A132252

Adjacent sequences: A007251 A007252 A007253 * A007255 A007256 A007257

Keyword

nonn

Author

N. J. A. Sloane

Status

approved