A063505 Number of n X n upper triangular binary matrices over GF(2) B such that B^2 = 0.

2, 8, 32, 320, 2592, 57472, 946176, 44302336, 1482686464, 143210315776, 9732400087040, 1915349322694656, 263918421714927616, 105091512697853313024, 29316605112733216538624, 23522116026027393322844160, 13266245323073952003913678848, 21392237922664971275489914126336, 24362629720999005014327927695736832

Offset: 2

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 30 2001

Keywords

• nonn

Comments

In the reference a more general formula is given for the number of such matrices over GF(q) for any q.

Links

• Robert Israel, Table of n, a(n) for n = 2..113
• Shalosh B. Ekhad, Doron Zeilberger, An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field, arXiv:math/9512224 [math.CO], 1995.
• Shalosh B. Ekhad, Doron Zeilberger, An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field, Elec. J. Comb. 3(1)(1996).

Crossrefs

Cf. A053722.

Programs

Maple

feven:= n -> add((binomial(2*n,n-3*j) - binomial(2*n,n-3*j-1))*2^(n^2-3*j^2-j),j=0..n/3):
fodd:= n -> add((binomial(2*n+1,n-3*j)-binomial(2*n+1,n-3*j-1))*2^(n^2+n-3*j^2-2*j),j=0..n/3):
seq(op([feven(i),fodd(i)]),i=1..20); # Robert Israel, Mar 01 2017

Mathematica

a[n_] := Sum[If[EvenQ[n], (Binomial[n, n/2 - 3j] - Binomial[n, n/2 - 3j - 1])*2^((n/2)^2 - 3j^2 - j), (Binomial[n, (n-1)/2 - 3j] - Binomial[n, (n-1)/2 - 3j - 1])*2^(((n-1)/2)^2 + (n-1)/2 - 3j^2 - 2j)], {j, 0, n/3}];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Sep 18 2018 *)

Formula

a(2n) = Sum_{j>=0} (C(2n, n - 3j) - C(2n, n - 3j - 1)) * 2^(n^2 - 3j^2 - j).

a(2n+1) = Sum_{j>=0} (C(2n + 1, n - 3j) - C(2n + 1, n - 3j - 1)) * 2^(n^2 + n - 3j^2 - 2j)

Extensions

More terms from Vladeta Jovovic, Aug 01 2001

Edited and more terms added by Robert Israel, Mar 01 2017