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%I #17 Sep 18 2018 04:15:25
%S 2,8,32,320,2592,57472,946176,44302336,1482686464,143210315776,
%T 9732400087040,1915349322694656,263918421714927616,
%U 105091512697853313024,29316605112733216538624,23522116026027393322844160,13266245323073952003913678848,21392237922664971275489914126336,24362629720999005014327927695736832
%N Number of n X n upper triangular binary matrices over GF(2) B such that B^2 = 0.
%C In the reference a more general formula is given for the number of such matrices over GF(q) for any q.
%H Robert Israel, <a href="/A063505/b063505.txt">Table of n, a(n) for n = 2..113</a>
%H Shalosh B. Ekhad, Doron Zeilberger, <a href="https://arxiv.org/abs/math/9512224">An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field</a>, arXiv:math/9512224 [math.CO], 1995.
%H Shalosh B. Ekhad, Doron Zeilberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r2">An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field</a>, Elec. J. Comb. 3(1)(1996).
%F a(2n) = Sum_{j>=0} (C(2n, n - 3j) - C(2n, n - 3j - 1)) * 2^(n^2 - 3j^2 - j).
%F a(2n+1) = Sum_{j>=0} (C(2n + 1, n - 3j) - C(2n + 1, n - 3j - 1)) * 2^(n^2 + n - 3j^2 - 2j)
%p 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):
%p 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):
%p seq(op([feven(i),fodd(i)]),i=1..20); # _Robert Israel_, Mar 01 2017
%t 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}];
%t Table[a[n], {n, 2, 20}] (* _Jean-François Alcover_, Sep 18 2018 *)
%Y Cf. A053722.
%K nonn
%O 2,1
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 30 2001
%E More terms from _Vladeta Jovovic_, Aug 01 2001
%E Edited and more terms added by _Robert Israel_, Mar 01 2017
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