A026105 Triangle T read by rows: differences of Motzkin triangle (A026300).

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 6, 7, 5, 1, 4, 10, 16, 18, 12, 1, 5, 15, 30, 44, 46, 30, 1, 6, 21, 50, 89, 120, 120, 76, 1, 7, 28, 77, 160, 259, 329, 316, 196, 1, 8, 36, 112, 265, 496, 748, 904, 841, 512, 1, 9, 45, 156, 413, 873, 1509, 2148, 2493, 2257, 1353, 1, 10, 55, 210, 614, 1442, 2795, 4530, 6150, 6898, 6103, 3610

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

For n >= 2, T(n,k)= number of nonnegative integer strings s(0),...,s(n) such that s(n)=n-k, s(0)=s(1)=1, |s(i)-s(i-1)|<=1 for i >= 2.

Examples

			1
1,1
1,1,1
1,2,3,2
1,3,6,7,5
1,4,10,16,18,12
1,5,15,30,44,46,30
		

Crossrefs

Right-hand columns include A002026, A026107, A026134, A026109, A026110.

Row sums are in A025566. Central column is in A026112.

Formula

T(n, k) = A026300(n, k) - A026300(n-1, k-1), T(1, 1) = 1.

T(i, 0)=1 for i >= 0, T(2, 1)=1, T(2, 2)=1, T(3, 1)=2, T(3, 2)=3, T(3, 3)=2; and for i >= 4, T(i, 1)=i-1, T(i, i)=T(i-1, i-2)+T(i-1, i-1) and T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i-1.

Right-hand columns have g.f. (1-z)*M^k, where M is g.f. of Motzkin numbers (A001006).

Extensions

Edited by Ralf Stephan, Dec 18 2004

a(65) corrected and more terms from Sean A. Irvine, Sep 16 2019

Offset set to 0 by Alois P. Heinz, Sep 16 2019