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A078805 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, every runlength of 1's odd and initial letter 0.

%I A078805 #14 Jul 23 2025 17:18:18
%S A078805 1,1,1,1,2,0,1,3,1,1,1,4,3,2,0,1,5,6,4,2,1,1,6,10,8,6,2,0,1,7,15,15,
%T A078805 13,6,3,1,1,8,21,26,25,16,9,2,0,1,9,28,42,45,36,22,9,4,1,1,10,36,64,
%U A078805 77,72,50,28,12,2,0,1,11,45,93,126,133,106,70,34,13,5,1,1,12,55,130,198,232
%N A078805 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, every runlength of 1's odd and initial letter 0.
%D A078805 Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.
%F A078805 T(n, k)=T(n-2, k)+T(n-2, k-1)+T(n-2, k-2)+T(n-3, k-1)-T(n-4, k-2) for 0<=k<=n, n>=4. (All numbers T(i, j) not in the array are 0, by definition of T.)
%e A078805 T(5,2) counts the words 01010, 01001, 00101.
%e A078805 Top of triangle T:
%e A078805   1 = T(1,0)
%e A078805   1 1 = T(2,0) T(2,1)
%e A078805   1 2 0 = T(3,0) T(3,1) T(3,2)
%e A078805   1 3 1 1
%e A078805   1 4 3 2 0
%t A078805 Clear[t];
%t A078805 t[n_,k_]:=0/;Not[0<=n<=200&&0<=k<=200];
%t A078805 t[1,0]=1;t[2,0]=1;t[2,1]=1;
%t A078805 t[3,0]=1;t[3,1]=2;t[3,2]=0; t[4,2]=1;
%t A078805 t[n_,k_]:=t[n,k]=t[n-2,k]+t[n-2,k-1]+t[n-2,k-2]+t[n-3,k-1]-t[n-4,k-2]
%t A078805 u=Table[t[n,k],{n,1,12},{k,0,n-1}]
%t A078805 Grid[u] (* triangle *)
%t A078805 Flatten[u] (* sequence *)
%t A078805 (* _Clark Kimberling_ Jul 18 2025 *)
%Y A078805 Cf. A078804, A078806.
%Y A078805 Row sums: A028495.
%K A078805 nonn,tabl
%O A078805 1,5
%A A078805 _Clark Kimberling_, Dec 07 2002