A014115 Order of a certain Clifford group in dimension 2^n (the automorphism group of the Barnes-Wall lattice for n != 3).

2, 8, 1152, 2580480, 89181388800, 48126558103142400, 409825748158189771161600, 55428899652335313894424707072000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

Links

• A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
• G. E. Wall, On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms, Nagoya Math. J., 21 (1962), pp. 199-222.
• Index entries for sequences related to Barnes-Wall lattices.

Crossrefs

Agrees with A014116 except at n=3.

Cf. A001309, A003956, A100221.

Programs

Maple

2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);

Mathematica

a[n_] := 2^(n^2+n+1)*(2^n - 1) * Product[4^i - 1, {i, 1, n-1}]; a[0] = 2; Array[a, 8, 0] (* Amiram Eldar, Jul 07 2025 *)

Python

from math import prod
def A014115(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

Formula

From Amiram Eldar, Jul 07 2025: (Start)

a(n) = 2^(n^2+n+1) * (2^n-1) * Product_{i=1..n-1} (2^(2*i)-1).

a(n) ~ c * 2^(2*n^2+n+1), where c = A100221. (End)