Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6 |

School of Mathematics

Liaoning Normal University

Dalian, 116029

PR China

Sai-nan Zheng

School of Mathematical Sciences

Dalian University of Technology

Dalian, 116024

PR China

**Abstract:**

Define an infinite lower triangular matrix
*D*(*e*,*h*) = [*d*_{n,k}]_{n,k ≥ 0} by the
recurrence *d*_{0,0} = *d*_{1,0} = *d*_{1,1} = 1,
*d*_{n,k} =
*d*_{n-1,k-1} + *e**d*_{n-1,k} +
*h**d*_{n-2,k-1},
where *e*, *h* are both nonnegative and *d*_{n,k} = 0 unless *n* ≥ *k* ≥ 0. We call
*D*(*e*, *h*) the *Delannoy-like triangle*.
The entries *d*_{n,k} count lattice
paths from (0, 0) to (*n* - *k*, *k*)
using the steps (0, 1), (1, 0) and (1, 1)
with assigned weights 1, *e*, and *h*. Some well-known combinatorial
triangles are such matrices, including the Pascal triangle *D*(1, 0), the
Fibonacci triangle *D*(0, 1), and the Delannoy triangle *D*(1, 1).
Futhermore, the Schröder triangle and Catalan triangle also arise as
inverses of Delannoy-like triangles. Here we investigate the total
positivity of Delannoy-like triangles. In addition, we show that each
row and diagonal of Delannoy-like triangles are all PF sequences.

(Concerned with sequences A001850 A007318 A008288 A026729 A033184 A132372.)

Received June 29 2016; revised version received December 20 2016.
Published in *Journal of Integer Sequences*, December 26 2016.

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