cp's OEIS Frontend

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

%I A001309 #33 Jul 07 2025 03:31:56
%S A001309 2,16,2304,5160960,178362777600,96253116206284800,
%T A001309 819651496316379542323200,110857799304670627788849414144000,
%U A001309 238987988705420266773820308079698247680000
%N A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.
%H A001309 A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, <a href="https://arxiv.org/abs/quant-ph/9608006">Quantum error correction via codes over GF(4)</a>, arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
%H A001309 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A001309 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0001038">The invariants of the Clifford groups</a>, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
%H A001309 <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>.
%H A001309 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>.
%F A001309 From _Amiram Eldar_, Jul 06 2025: (Start)
%F A001309 a(n) = 2^(n^2+n+2) * (2^n-1) * Product_{k=1..n-1} (2^(2*k)-1).
%F A001309 a(n) ~ c * 2^(2*n^2+n+2), where c = A100221. (End)
%p A001309 2^(n^2+n+2) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);
%t A001309 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 *)
%o A001309 (Python)
%o A001309 from math import prod
%o A001309 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
%Y A001309 2^(2n+2) times order of Chevalley group D_n (2) (cf. A001308). Twice A014115. See also A014116, A003956 (for the complex group).
%Y A001309 Cf. A100221.
%K A001309 nonn,easy,nice
%O A001309 0,1
%A A001309 _N. J. A. Sloane_, _Peter Shor_