A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.

2, 16, 2304, 5160960, 178362777600, 96253116206284800, 819651496316379542323200, 110857799304670627788849414144000, 238987988705420266773820308079698247680000

Offset: 0

Author

N. J. A. Sloane, Peter Shor

Keywords

• nonn
• easy
• nice

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. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• G. Nebe, E. M. Rains and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
• Index entries for sequences related to Barnes-Wall lattices.
• Index entries for sequences related to groups.

Crossrefs

2^(2n+2) times order of Chevalley group D_n (2) (cf. A001308). Twice A014115. See also A014116, A003956 (for the complex group).

Cf. A100221.

Programs

Maple

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

Mathematica

a[0] = 2; a[n_] := 2^(n^2+n+2) * (2^n-1) * Product[2^(2*i)-1, {i, 1, n-1}]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jul 16 2015, after Maple *)

Python

from math import prod
def A001309(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

Formula

From Amiram Eldar, Jul 06 2025: (Start)

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

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