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A368745
Triangular array read by rows: T(n, k) is the number of n X 2 Young tableaux with k vertical walls.
1
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
0,3
COMMENTS
Same as A336524 with the main diagonal removed.
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.
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
EXAMPLE
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,
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 *)
CROSSREFS
Cf. A336524.
Sequence in context: A173869 A247377 A167909 * A185421 A219976 A063944
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Bala, Feb 04 2024
STATUS
approved