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A063505
Number of n X n upper triangular binary matrices over GF(2) B such that B^2 = 0.
2
2, 8, 32, 320, 2592, 57472, 946176, 44302336, 1482686464, 143210315776, 9732400087040, 1915349322694656, 263918421714927616, 105091512697853313024, 29316605112733216538624, 23522116026027393322844160, 13266245323073952003913678848, 21392237922664971275489914126336, 24362629720999005014327927695736832
OFFSET
2,1
COMMENTS
In the reference a more general formula is given for the number of such matrices over GF(q) for any q.
LINKS
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.
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)
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 *)
CROSSREFS
Cf. A053722.
Sequence in context: A369645 A134751 A139014 * A085466 A084039 A135620
KEYWORD
nonn
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 30 2001
EXTENSIONS
More terms from Vladeta Jovovic, Aug 01 2001
Edited and more terms added by Robert Israel, Mar 01 2017
STATUS
approved