%I #15 Jun 21 2022 05:07:23

%S 2,8,1152,696729600,89181388800,48126558103142400,

%T 409825748158189771161600,55428899652335313894424707072000

%N Order of automorphism group of Barnes-Wall lattice in dimension 2^n.

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

%H 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 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 <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>

%p 2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1); # except for n=3.

%o (Python)

%o from math import prod

%o def A014116(n): return 2+696729598*(n//3) if n == 0 or n == 3 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 Agrees with A014115 except at n=3. Equals half of A001309. Cf. A003956.

%K nonn

%O 0,1

%A _N. J. A. Sloane_