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A091325 Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes (n >= 1, 0 <= k <= n-1).

%I A091325 #16 Jul 01 2025 16:17:51
%S A091325 1,1,1,1,1,1,1,2,2,1,1,2,3,2,1,1,3,5,5,3,1,1,3,7,9,7,3,1,1,4,10,17,17,
%T A091325 10,4,1,1,4,13,26,35,26,13,4,1
%N A091325 Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes (n >= 1, 0 <= k <= n-1).
%C A091325 "Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.
%H A091325 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 A091325 <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>
%F A091325 T(n, 0) = T(n, n-1) = 1, T(n, n) = 0; T(n, 1) = floor(n/2); T(n, k) = T(n, n-k-1).
%e A091325 Triangle begins
%e A091325 1
%e A091325 1 1
%e A091325 1 1 1
%e A091325 1 2 2 1
%e A091325 1 2 3 2 1
%e A091325 1 3 5 5 3 1
%o A091325 (Magma) P<t> := PolynomialAlgebra(Rationals()); qbinom := function(n,k) return &*[Rationals()|(1-2^(n+1-i))/(1-2^i):i in [1..k]]; end function;
%o A091325 for n in [2..9] do G := Sym(n); refmod := PermutationModule(G,GF(2)); refmod := refmod/sub<refmod|[1:i in [1..n]]>; CL := ConjugacyClasses(G); acc := &+[qbinom(n-1,k)*t^k:k in [0..n-1]]; n,(acc+&+[P|c[2]*&+[t^(n-1-Dimension(s)):s in Submodules(Restriction(refmod,sub<G|c[3]>))]:c in CL|c[1] ne 1])/#G; end for;
%Y A091325 Row sums give A091326.
%K A091325 nonn,tabl,more
%O A091325 1,8
%A A091325 _N. J. A. Sloane_, Mar 01 2004
%E A091325 Rows 7 - 9 computed by Eric Rains (rains(AT)caltech.edu) using MAGMA, Mar 01 2004
%E A091325 It would be nice even to have a continuation of the numbers for dimension 2, T(n,2).