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A063505
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Number of n X n upper triangular binary matrices over GF(2) B such that B^2 = 0.
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2
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2, 8, 32, 320, 2592, 57472, 946176, 44302336, 1482686464, 143210315776, 9732400087040, 1915349322694656, 263918421714927616, 105091512697853313024, 29316605112733216538624, 23522116026027393322844160, 13266245323073952003913678848, 21392237922664971275489914126336, 24362629720999005014327927695736832
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OFFSET
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2,1
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COMMENTS
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In the reference a more general formula is given for the number of such matrices over GF(q) for any q.
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LINKS
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FORMULA
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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)
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MAPLE
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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
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 30 2001
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EXTENSIONS
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STATUS
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approved
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