OFFSET
1,8
COMMENTS
"Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.
LINKS
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
FORMULA
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).
EXAMPLE
Triangle begins
1
1 1
1 1 1
1 2 2 1
1 2 3 2 1
1 3 5 5 3 1
PROG
(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;
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;
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 01 2004
EXTENSIONS
Rows 7 - 9 computed by Eric Rains (rains(AT)caltech.edu) using MAGMA, Mar 01, 2004
It would be nice even to have a continuation of the numbers for dimension 2, T(n,2).
STATUS
approved