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

0,8

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.

Links

Table of n, a(n) for n=0..77.

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).

Example

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.

Sequence in context: A182457 A368605 A334715 * A303868 A060475 A168069

Adjacent sequences: A026102 A026103 A026104 * A026106 A026107 A026108

Keyword

nonn,tabl

Author

Clark Kimberling

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

Status

approved