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A214776 Number A(n,k) of standard Young tableaux of shape [n*k,n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A214776 #39 Dec 16 2020 17:54:22
%S A214776 1,1,0,1,1,0,1,2,2,0,1,3,9,5,0,1,4,20,48,14,0,1,5,35,154,275,42,0,1,6,
%T A214776 54,350,1260,1638,132,0,1,7,77,663,3705,10659,9996,429,0,1,8,104,1120,
%U A214776 8602,40480,92092,62016,1430,0,1,9,135,1748,17199,115101,451269,807300,389367,4862,0
%N A214776 Number A(n,k) of standard Young tableaux of shape [n*k,n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A214776 A(n,k) is also the number of binary words with n*k 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's.  The A(2,2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.
%H A214776 Alois P. Heinz, <a href="/A214776/b214776.txt">Antidiagonals n = 0..140</a>
%H A214776 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Barry/barry444.html">On the Central Antecedents of Integer (and Other) Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
%H A214776 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A214776 A(n,k) = max(0, C((k+1)*n,n)*((k-1)*n+1)/(k*n+1)).
%e A214776 Square array A(n,k) begins:
%e A214776   1,   1,    1,     1,     1,      1,      1, ...
%e A214776   0,   1,    2,     3,     4,      5,      6, ...
%e A214776   0,   2,    9,    20,    35,     54,     77, ...
%e A214776   0,   5,   48,   154,   350,    663,   1120, ...
%e A214776   0,  14,  275,  1260,  3705,   8602,  17199, ...
%e A214776   0,  42, 1638, 10659, 40480, 115101, 272272, ...
%p A214776 A:= (n, k)-> max(0, binomial((k+1)*n, n)*((k-1)*n+1)/(k*n+1)):
%p A214776 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A214776 a[n_, k_] := Max[0, Binomial[(k+1)*n, n]*((k-1)*n+1)/(k*n+1)]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 01 2013, after Maple *)
%Y A214776 Columns k=0-10 give: A000007, A000108, A174687, A126596, A215541, A215542, A215551, A215552, A215553, A215554, A215555.
%Y A214776 Rows n=0-10 give: A000012, A001477, A014107(k+1), A215543, A215544, A215545, A215546, A215547, A215548, A215549, A215550.
%Y A214776 Main diagonal gives: A215557.
%K A214776 nonn,tabl
%O A214776 0,8
%A A214776 _Alois P. Heinz_, Jul 28 2012