cp's OEIS Frontend

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

%I A014115 #22 Jul 07 2025 03:32:04
%S A014115 2,8,1152,2580480,89181388800,48126558103142400,
%T A014115 409825748158189771161600,55428899652335313894424707072000
%N A014115 Order of a certain Clifford group in dimension 2^n (the automorphism group of the Barnes-Wall lattice for n != 3).
%D A014115 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.
%H A014115 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 A014115 G. E. Wall, <a href="https://doi.org/10.1017/S0027763000023837">On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms</a>, Nagoya Math. J., 21 (1962), pp. 199-222.
%H A014115 <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>.
%F A014115 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A014115 a(n) = 2^(n^2+n+1) * (2^n-1) * Product_{i=1..n-1} (2^(2*i)-1).
%F A014115 a(n) ~ c * 2^(2*n^2+n+1), where c = A100221. (End)
%p A014115 2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);
%t A014115 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 *)
%o A014115 (Python)
%o A014115 from math import prod
%o A014115 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
%Y A014115 Agrees with A014116 except at n=3.
%Y A014115 Cf. A001309, A003956, A100221.
%K A014115 nonn
%O A014115 0,1
%A A014115 _N. J. A. Sloane_