cp's OEIS Frontend

Diagonal of rational function R(x,y,z,t) = 1/(1 + y + z + x*y + y*z + t*x*z + (t+1)*x*y*z) with respect to x, y, z, i.e., T(n,k) = [(xyz)^n*t^k] R(x,y,z,t). - Gheorghe Coserea, Jul 01 2018

Also T(n,k) = binomial(n-1, k-1)*binomial(n, k-1), related to Narayana polynomials (see Sulanke reference). - Roger L. Bagula, Apr 09 2008

h-vector for cluster complex associated to the root system B_n. See p. 8, Athanasiadis and C. Savvidou. - Tom Copeland, Oct 19 2014

Definition of the Hilbert transform of a triangular array:

For many square arrays in the database the entries in a row are polynomial in the column index, of degree d say and hence the row generating function has the form P(x)/(1-x)^(d+1), where P is some polynomial function. Often the array whose rows are formed from the coefficients of these P polynomials is of independent interest. This suggests the following definition.

Let [L(n,k)]n,k>=0 be a lower triangular array and let R(n,x) := sum {k = 0 .. n} L(n,k)*x^k, denote the n-th row generating polynomial of L. Then we define the Hilbert transform of L, denoted Hilb(L), to be the square array whose n-th row, n >= 0, has the generating function R(n,x)/(1-x)^(n+1).

In this particular case, L is the array A060187, the array of Eulerian numbers of type B, whose row polynomials are the h-polynomials for permutohedra of type B. The Hilbert transform is an infinite Vandermonde matrix V(1,3,5,...).

We illustrate the Hilbert transform with a few examples:

(1) The Delannoy number array A008288 is the Hilbert transform of Pascal's triangle A007318 (view as the array of coefficients of h-polynomials of n-dimensional cross polytopes).

(2) The transpose of the array of nexus numbers A047969 is the Hilbert transform of the triangle of Eulerian numbers A008292 (best viewed in this context as the coefficients of h-polynomials of n-dimensional permutohedra of type A).

(3) The sequence of Eulerian polynomials begins [1, x, x + x^2, x + 4*x^2 + x^3, ...]. The coefficients of these polynomials are recorded in triangle A123125, whose Hilbert transform is A004248 read as square array.

(4) A108625, the array of crystal ball sequences for the A_n lattices, is the Hilbert transform of A008459 (viewed as the triangle of coefficients of h-polynomials of n-dimensional associahedra of type B).

(5) A142992, the array of crystal ball sequences for the C_n lattices, is the Hilbert transform of A086645, the array of h-vectors for type C root polytopes.

(6) A108553, the array of crystal ball sequences for the D_n lattices, is the Hilbert transform of A108558, the array of h-vectors for type D root polytopes.

(7) A086764, read as a square array, is the Hilbert transform of the rencontres numbers A008290.

(8) A143409 is the Hilbert transform of triangle A073107.

Let S_n denote the symmetric group on {1,2,...,n}. A permutation p_1p_2...p_n in S_n has a descent at position i (1 <= i <= n-1) if p_i > p_(i+1). The Eulerian numbers A008292 (with an offset of 0 in the column indexing) enumerate permutations by descents. We define a pair of permutations p_1p_2...p_n and q_1q_2...q_n to have a common descent at position i (1 <= i <= n-1) if both p_i > p_(i+1) and q_i > q_(i+1). For example, the permutations (3241) and (4231) in S_4 have common descents at positions i = 1 and i = 3. The table entry T(n,k) gives the number of pairs of permutations in the Cartesian product S_n x S_n with k common descents.

The generalized Stirling numbers associated with this triangle is A061691. See also A192722.

The first array in the series is Pascal's triangle, A007318. The initial partition for each subsequent array in the series is chosen as described in A053445. When cells are squared, as in A008459, row sums yield 1, 2, 6, 24, ... (A000142). E.g., (1 + 16 + 36 + 16 + 1) + (25 + 25) = 70 + 50 = 120 using row five from A007318 and row two from this array.

Number of lattice paths from (0,0) to (n,k) in exactly n+k+2 steps. Moves allowed: (0,1), (0,-1), (-1,0) and (1,0). Paths must stay in Quadrant I but may touch the axes. - Juan Luis Vargas-Molina, Mar 03 2014

Row n equals the (n+1)-th differences of row n of the square array A108553. G.f. of row n equals: (1-x)^(n+1)*CBD_n(x), where CBD_n denotes the g.f. of the crystal ball sequence for D_n lattice.

From Peter Bala, Oct 23 2008: (Start)

Let D_n be the root lattice generated as a monoid by {+-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(D_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(D_n) [Ardila et al.]. See A108556 for the corresponding array of f-vectors for these type D_n polytopes. See A008459 for the array of h-vectors for type A_n polytopes and A086645 for the array of h-vectors associated with type C_n polytopes.

The Hilbert transform of this array (as defined in A145905) equals A108553.

(End)

Row sums of the fourth power of A008459. - Peter Bala, Mar 05 2013

Row sums of the fifth power of A008459. - Peter Bala, Mar 05 2013

a(n)/6^(2n) is the probability that two throws of n 6-sided dice will give the same result - Henry Bottomley, Aug 30 2016

Since the triangle is symmetric, the probability that a one-dimensional random walk returns to the origin at all in the steps m through to 2m is 1/2 (for m odd).

Diagonal sums are A106183. - Paul Barry, Apr 24 2005

First column is A000984. Second column is A110609 = n^2*A000108. Row sums are A005809.