A078821 -id:A078821 - cp's OEIS frontend

A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 0, 0, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 0, 0, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 0, 0, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 0, 0

Offset: 1

Clark Kimberling, Dec 07 2002

Row sums: 1,1,2,3,5,8,13,..., the Fibonacci numbers (A000045).

			T(6,2) counts the words 010001 and 000101. Top of triangle:
1 = T(1,0)
0 1 = T(2,0) T(2,1)
1 1 0
0 1 1 1
1 2 1 1 0

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.

Cf. A078808, A078821, A079487, A123245, A077419.

T(n, k)=T(n-1, n-k-1)+T(n-3, n-k-3)+...+T(n-2m-1, n-k-2m-1), where m=[(n-1)/2] and (by definition) T(i, j)=0 if i<0 or j<0 or i=j.

Row 0 removed to stick to the triangle format by Andrey Zabolotskiy, Sep 22 2017

A078808 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 2, 2, 2, 1, 0, 0, 0, 1, 2, 3, 3, 3, 1, 0, 1, 3, 4, 5, 4, 3, 1, 0, 0, 0, 1, 3, 5, 7, 7, 6, 4, 1, 0, 1, 4, 7, 10, 11, 10, 7, 4, 1, 0, 0, 0, 1, 4, 8, 13, 16, 17, 14, 10, 5, 1, 0, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 0, 0, 0, 1, 5, 12

Offset: 0

Clark Kimberling, Dec 07 2002

Row sums: 0,1,1,2,3,5,8,13,..., the Fibonacci numbers (A000045).

			T(6,2) counts the words 101000 and 100010. Top of triangle:
0 = T(0,0)
0 1 = T(1,0) T(1,1)
0 1 0 = T(2,0) T(2,1) T(2,2)
0 0 1 1
0 1 1 1 0
0 0 1 1 2 1

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.

Cf. A078807, A078821.

T(n, k)=t(n, n, n+1-k), where t is the array given by A078807.

cp's OEIS Frontend

A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

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