A368745 Triangular array read by rows: T(n, k) is the number of n X 2 Young tableaux with k vertical walls.

1, 1, 2, 2, 6, 6, 5, 20, 30, 20, 14, 70, 140, 140, 70, 42, 252, 630, 840, 630, 252, 132, 924, 2772, 4620, 4620, 2772, 924, 429, 3432, 12012, 24024, 30030, 24024, 12012, 3432, 1430, 12870, 51480, 120120, 180180, 180180, 120120, 51480, 12870, 4862, 48620, 218790, 583440, 1021020

Offset: 0

Author

Peter Bala, Feb 04 2024

Keywords

• nonn
• tabl
• easy

Comments

Same as A336524 with the main diagonal removed.

Examples

			Triangle T(n, k) begins:
    1,
    1,    2,
    2,    6,     6,
    5,   20,    30,    20,
   14,   70,   140,   140,    70,
   42,  252,   630,   840,   630,   252,
  132,  924,  2772,  4620,  4620,  2772,  924,
  429, 3432, 12012, 24024, 30030, 24024, 12012, 3432,
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
• Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.

Crossrefs

Cf. A336524.

Programs

Maple

A368745 := (n, k) -> 1/(n + 1 - k)*binomial(n, k)*binomial(2*n, n):
seq(print(seq(A368745(n, k), k = 0..n)), n = 0..10);

Mathematica

A368745row[n_] := Binomial[n, #]*Binomial[2*n, n]/(n+1-#) & [Range[0, n]];
Array[A368745row, 10, 0] (* Paolo Xausa, Feb 27 2024 *)

Formula

T(n, k) = 1/(n + 1 - k)*binomial(n, k)*binomial(2*n, n).

G.f.: (sqrt(1 - 4*x*y) - sqrt(1 - 4*x*(1 + y)))/(2*x). - Stefano Spezia, Feb 04 2024