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Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's and all runlengths odd.
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%I #13 Mar 05 2024 07:10:28

%S 0,1,1,0,2,0,1,1,1,1,0,2,2,2,0,1,2,2,2,2,1,0,2,4,4,4,2,0,1,3,4,5,5,4,

%T 3,1,0,2,6,8,10,8,6,2,0,1,4,7,10,12,12,10,7,4,1,0,2,8,14,20,22,20,14,

%U 8,2,0,1,5,11,18,25,29,29,25,18,11,5,1,0,2,10,22,36,48,52,48,36,22,10,2,0

%N Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's and all runlengths odd.

%C Rows are symmetric. Row sums (0,2,2,4,6,10,16,26,...) are given by 2*F(n), where F(n) is the n-th Fibonacci number, A000045(n).

%D 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.

%H Zachary Greenberg, Dani Kaufman, and Anna Wienhard, <a href="https://arxiv.org/abs/2402.19300">SL_2-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers</a>, arXiv:2402.19300 [math.RA], 2024. See p. 38.

%F T(n, k) = A078807(n, k) + A078808(n, k).

%e T(6,2) counts the words 010001, 000101, 101000 and 100010.

%e Top of triangle:

%e 0

%e 1 1

%e 0 2 0

%e 1 1 1 1

%e 0 2 2 2 0

%e 1 2 2 2 2 1

%Y Cf. A000045, A078807, A078808.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Dec 07 2002