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

0,1

References

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

Links

Table of n, a(n) for n=0..7.

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.

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)

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 *)

Prog

(Python)
from math import prod
def A014115(n): return 2 if n == 0 else ((1<<n)-1)*prod((1<<i)-1 for i in range(2, 2*n-1, 2)) << n*(n+1)+1 # Chai Wah Wu, Jun 20 2022

Crossrefs

Agrees with A014116 except at n=3.

Cf. A001309, A003956, A100221.

Sequence in context: A061591 A103085 A084148 * A014116 A027668 A121015

Adjacent sequences: A014112 A014113 A014114 * A014116 A014117 A014118

Keyword

nonn

Author

N. J. A. Sloane

Status

approved