%I #15 Nov 17 2018 15:33:04
%S 2,3,4,4,10,8,5,20,35,16,6,35,112,126,32,7,56,294,672,462,64,8,84,672,
%T 2772,4224,1716,128,9,120,1386,9504,28314,27456,6435,256,10,165,2640,
%U 28314,151008,306735,183040,24310,512,11,220,4719,75504,674817
%N Square array T(n,k) read by antidiagonals: T(n,k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1).
%C Number of semistandard Young tableaux with at most n columns and with entries in [k].
%C T(n,k) is the number of k X k symmetric matrices with entries in 0..n with each row (and column) in nondecreasing order. - _R. H. Hardin_, Jul 08 2008
%H M. Lederer, <a href="http://arXiv.org/abs/math.CO/0501132">A determinant-like formula for the Kostka numbers</a>
%F It appears that T is identical to the reflected triangle A073165, i.e. T(n, k) = Prod[i=1..floor((k+1)/2), C(n+k+2i-1-(k mod 2), 4i-1-2(k mod 2))] / Prod[i=0..floor((k-1)/2), C(2k-2i-1, 2i)].
%e Square array T(n,k) begins:
%e 2, 4, 8, 16, 32, 64, ...
%e 3, 10, 35, 126, 462, 1716, ...
%e 4, 20, 112, 672, 4224, 27456, ...
%e 5, 35, 294, 2772, 28314, 306735, ...
%e 6, 56, 672, 9504, 151008, 2617472, ...
%e 7, 84, 1386, 28314, 674817, 18076916, ...
%e ...
%t T[n_, k_] := Product[(n + i + j - 1)/(i + j - 1), {i, 1, k}, {j, i, k}];
%t Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 06 2018 *)
%Y Rows include A000079, A001700, A003645, A000356.
%Y Main diagonal is A049505.
%K nonn,tabl
%O 1,1
%A _Ralf Stephan_, Jan 14 2005