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The old name was: Number of closed walks of 2n unit steps north, east, south, or west starting and ending at the origin and confined to the first octant.

Image of Catalan numbers (A000108) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

The Niederhausen reference counts various classes of first octant paths by number of contacts with the line y=x. - David Callan, Sep 18 2007

In Sloane and Plouffe (1995) this was incorrectly described as "Dyck paths".

Also matchings avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013

From Bruce Westbury, Aug 22 2013: (Start)

a(n) is also the number of nested pairs of Dyck paths of length n starting and ending at the origin;

a(n) is also the number of 3-noncrossing perfect matchings on 2n points;

a(n) is also the number of 2-triangulations on n-gon;

a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the spin representation of Spin(5);

a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the defining representation of Sp(4). (End)

a(-1) = -3/2, a(-2) = -1/4 makes some formulas true for all n in Z. - Michael Somos, Oct 02 2014

a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the pattern 312. - Colin Defant, Jun 08 2019

As square array: number of certain symmetric plane partitions, see Forrester/Gamburd paper.

Formatted as a square array, the column k gives the Hankel transform of the Catalan numbers (A000108) beginning at A000108(k); example: Hankel transform of [42, 132, 429, 1430, 4862, ...] is [42, 594, 4719, 26026, 111384, ...] (see A091962). - Philippe Deléham, Apr 12 2007

As square array T(n,k): number of all k-watermelons with a wall of length n. - Ralf Stephan, May 09 2007

Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used." de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." It turns out that for the current array, T(n,k) = b(n-k,k) for n >= 0 and 0 <= k <= n. - Petros Hadjicostas, Sep 04 2019

As square array, b(k, n) = T(n+k-1, n) for k >= 1 and n >= 1 is the number of n-tuples P = (p_1, p_2, ..., p_n) of non-intersecting lattice paths that lie below the diagonal, such that each p_i starts at (i, i) and ends at (2n+k-i, 2n+k-i). (This is just a different way of looking at n-watermelons with a wall of length k since many of the steps of these paths are going to be fixed while the rest form an n-watermelon. See the Krattenthaler et al. paper.) Equivalently b(k, n) is the number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths, each with 2k steps such that for every i (1 <= i <= n-1), p_i is included in p_{i+1}. A Dyck path p is said to be included in a Dyck path q if the height of path p after j steps is at most the height of path q after j steps, for all j (1 <= j <= 2k). - Farzan Byramji, Jun 17 2021

There is another version in A078920. - Philippe Deléham, Apr 12 2007 [In other words, T(n,k) = A078920(n,n-k). - Petros Hadjicostas, Oct 19 2019]

This array is a variant of the triangles A078920 and A123352 extended to the trivial cases (here for k=0).

a(n) is the determinant of the 3 X 3 Hankel matrix [a_0, a_1, a_2 ; a_1, a_2, a_3 ; a_2, a_3, a_4] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007

Third subdiagonal in A123352, equivalent to the 6th subdiagonal in A185249, its "aerated" version with additional subdiagonals entirely filled with zeros. - R. J. Mathar, Feb 18 2011

a(n) is the determinant of the 5 X 5 Hankel matrix [a_0, a_1, a_2, a_3, a_4 ; a_1, a_2, a_3, a_4, a_5 ; a_2, a_3, a_4, a_5, a_6 ; a_3, a_4, a_5, a_6, a_7 ; a_4, a_5, a_6, a_7, a_8] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007

I have a photocopy of certain pages of the thesis, but unfortunately not enough to find the definition of this table. I have written to the author.

(Added later) However, Alois P. Heinz found a formula involving Catalan numbers which matches all the data and is surely correct, so the triangle is no longer a mystery.

Reading upwards along antidiagonals gives A123352.

From Petros Hadjicostas, Sep 04 2019: (Start)

Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used."

de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." In Section 6 of their paper, they give an interpretation of this formula in terms of Pfaffians and perfect matchings.

It turns out that for the current array, T(n,k) = b_{k, (n-k)/2} if n-k is even, and 0 otherwise (for n >= 0 and 0 <= k <= n). It is unknown, however, what kind of interpretation Myriam de Sainte-Catherine gave to the number T(n,k) three years earlier in her 1983 Ph.D. dissertation. It may be distantly related to the numbers b_{n,k} that are found in her 1986 paper with G. Viennot.

(End)

The T(n, k) for n and k same parity are the numbers in the upper triangle of the Catalan Number Wall in "Number Walls in Combinatorics". Thus 0 = T(n-1, k+1)*T(n+1, k-1) - T(n-1, k-1)*T(n+1, k+1) + T(n, k)^2 for all n, k. - Michael Somos, Aug 15 2023