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See related triangle A138158.

Row sums are the Catalan numbers (A000108), set q=1 in the g.f. to see this.

Antidiagonal sums equal A005169, the number of fountains of n coins.

The maximum in each row of the triangle is A274291. - Torsten Muetze, Nov 28 2018

The area between a Dyck path and the x-axis may be decomposed into unit area triangles of two types - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, y+1) and (x+1, y+1). The table entry T(n,k) equals the number of Dyck paths of semilength n containing k down triangles. See the illustration in the Links section. Cf. A239927. - Peter Bala, Jul 11 2019

The row polynomials of this table are a q-analog of the Catalan numbers due to Carlitz and Riordan. For MacMahon's q-analog of the Catalan numbers see A129175. - Peter Bala, Feb 28 2023

Column sums of A129182.

From Peter Bala, Jul 29 2019: (Start)

a(n) = the number of triangle stacks of large Schröder type on n triangles. See Links for a definition and an illustration.

Cf. A224704, which enumerates triangle stacks (of small Schröder type) on n triangles and A143951, which enumerates triangle stacks (of Dyck type) on n triangles. (End)

From Peter Bala, Jul 29 2019: (Start)

a(n) is the number of triangle stacks of large Schröder type containing n down-triangles. See Links for a definition and an illustration.

Cf. A088352 for triangle stacks of large Schröder type on n triangles. Cf. A224704, which enumerates triangle stacks (of small Schröder type) on n triangles. (End)

A239927 is the companion triangle for Dyck paths.

A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.

The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles are of two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). A small Schröder path of semilength k has k up-triangles in the bottom row of its stack. See the illustration in the Links section for an example. Thus an alternative description of the triangle entry T(n,k) is the number of n triangle stacks, in the sense of A224704, containing k up-triangles in the bottom row.

A227543 is the companion triangle for Dyck paths.

Number of n triangle stacks, in the sense of A224704, containing k down- triangles.

A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.

The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). See the illustration in the Links section for an example.

A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.

The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). These are the triangle stacks of A224704. Here we enumerate triangle stacks with n >= 1 up-triangles in the bottom row of the stack (corresponding to small Schröder paths of semilength n) and containing k >= 0 down-triangles in the stack. See the illustration in the Links section for an example.

We define two types of plane triangles of unit area - up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1).

To construct a triangle stack of large Schröder type we start with a horizontal row of k contiguous down-triangles forming the base row of the stack. Subsequent rows of the stack are formed by placing up-triangles on some, all or none of the down-triangles of the previous row. In the spaces between pairs of adjacent up-triangles further down-triangles may be placed. For an example, see the illustration in the Links section. There is an obvious bijective correspondence between triangle stacks of large Schröder type with a base of k down-triangles and large Schröder paths of semilength k. For another version of this array see A129179.

For triangle stacks of small Schröder type, where the base row consists of contiguous up-triangles, see A224704.