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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.

%I A368745 #17 Feb 27 2024 06:50:36
%S A368745 1,1,2,2,6,6,5,20,30,20,14,70,140,140,70,42,252,630,840,630,252,132,
%T A368745 924,2772,4620,4620,2772,924,429,3432,12012,24024,30030,24024,12012,
%U A368745 3432,1430,12870,51480,120120,180180,180180,120120,51480,12870,4862,48620,218790,583440,1021020
%N A368745 Triangular array read by rows: T(n, k) is the number of n X 2 Young tableaux with k vertical walls.
%C A368745 Same as A336524 with the main diagonal removed.
%H A368745 Paolo Xausa, <a href="/A368745/b368745.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H A368745 Cyril Banderier, Philippe Marchal, and Michael Wallner, <a href="https://arxiv.org/abs/1805.09017">Rectangular Young tableaux with local decreases and the density method for uniform random generation</a> (short version), arXiv:1805.09017 [cs.DM], 2018.
%F A368745 T(n, k) = 1/(n + 1 - k)*binomial(n, k)*binomial(2*n, n).
%F A368745 G.f.: (sqrt(1 - 4*x*y) - sqrt(1 - 4*x*(1 + y)))/(2*x). - _Stefano Spezia_, Feb 04 2024
%e A368745 Triangle T(n, k) begins:
%e A368745     1,
%e A368745     1,    2,
%e A368745     2,    6,     6,
%e A368745     5,   20,    30,    20,
%e A368745    14,   70,   140,   140,    70,
%e A368745    42,  252,   630,   840,   630,   252,
%e A368745   132,  924,  2772,  4620,  4620,  2772,  924,
%e A368745   429, 3432, 12012, 24024, 30030, 24024, 12012, 3432,
%p A368745 A368745 := (n, k) -> 1/(n + 1 - k)*binomial(n, k)*binomial(2*n, n):
%p A368745 seq(print(seq(A368745(n, k), k = 0..n)), n = 0..10);
%t A368745 A368745row[n_] := Binomial[n, #]*Binomial[2*n, n]/(n+1-#) & [Range[0, n]];
%t A368745 Array[A368745row, 10, 0] (* _Paolo Xausa_, Feb 27 2024 *)
%Y A368745 Cf. A336524.
%K A368745 nonn,tabl,easy
%O A368745 0,3
%A A368745 _Peter Bala_, Feb 04 2024