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

0,1

Links

Table of n, a(n) for n=0..8.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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, 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

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 *)

Prog

(Python)
from math import prod
def A001309(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)+2 # Chai Wah Wu, Jun 20 2022

Crossrefs

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

Sequence in context: A091479 A016031 A331691 * A132569 A165644 A338005

Adjacent sequences: A001306 A001307 A001308 * A001310 A001311 A001312

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Peter Shor

Status

approved