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A099766 Triangle read by rows: T(n,k) = number of unbordered binary words of length n and weight k, n >= 0, 0 <= k <= n.

%I A099766 #9 Jul 17 2020 00:33:49
%S A099766 1,1,1,0,2,0,0,2,2,0,0,2,2,2,0,0,2,4,4,2,0,0,2,4,8,4,2,0,0,2,6,12,12,
%T A099766 6,2,0,0,2,6,18,22,18,6,2,0,0,2,8,24,40,40,24,8,2,0,0,2,8,32,60,80,60,
%U A099766 32,8,2,0,0,2,10,40,92,140,140,92,40,10,2,0,0,2,10,50,128,232
%N A099766 Triangle read by rows: T(n,k) = number of unbordered binary words of length n and weight k, n >= 0, 0 <= k <= n.
%H A099766 T. Harju and D. Nowotka, <a href="http://oldtucs.abo.fi/publications/attachment.php?fname=TR630.pdf">Counting bordered and primitive words with a fixed weight</a>, TUCS Technical Report, No 630, Turku, November 2004. [This is the triangle U(n,k).]
%H A099766 T. Harju and D. Nowotka, <a href="https://doi.org/10.1016/j.tcs.2005.03.040">Counting bordered and primitive words with a fixed weight</a>, Theoret. Comput. Sci. 340 (2005), no. 2, 273-279. [This is the triangle U(n,k).]
%F A099766 See Maple code.
%e A099766 Triangle begins:
%e A099766 .1
%e A099766 .1,1
%e A099766 .0,2,0
%e A099766 .0,2,2,0
%e A099766 .0,2,2,2,0
%e A099766 .0,2,4,4,2,0
%e A099766 .0,2,4,8,4,2,0
%p A099766 U:=proc(n,k) option remember; if n < 1 then RETURN(0); fi; if n = 1 then RETURN(1); fi; if n > 1 and k = 0 then RETURN(0); fi; if k > 1 and k >= n then RETURN(0); fi; U(n-1,k)+U(n-1,k-1)-E(n,k); end;
%p A099766 E:=proc(n,k) option remember; if n mod 2 = 0 and k mod 2 = 0 then U(n/2,k/2) else 0; fi; end;
%Y A099766 Row sums give A003000. Cf. A099768, A102416.
%K A099766 nonn,tabl
%O A099766 0,5
%A A099766 _N. J. A. Sloane_, Nov 11 2004