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A102539 Square array T(n,k) read by antidiagonals: T(n,k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1).

%I A102539 #15 Nov 17 2018 15:33:04
%S A102539 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 A102539 2772,4224,1716,128,9,120,1386,9504,28314,27456,6435,256,10,165,2640,
%U A102539 28314,151008,306735,183040,24310,512,11,220,4719,75504,674817
%N A102539 Square array T(n,k) read by antidiagonals: T(n,k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1).
%C A102539 Number of semistandard Young tableaux with at most n columns and with entries in [k].
%C A102539 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 A102539 M. Lederer, <a href="http://arXiv.org/abs/math.CO/0501132">A determinant-like formula for the Kostka numbers</a>
%F A102539 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 A102539 Square array T(n,k) begins:
%e A102539   2,  4,    8,    16,     32,       64, ...
%e A102539   3, 10,   35,   126,    462,     1716, ...
%e A102539   4, 20,  112,   672,   4224,    27456, ...
%e A102539   5, 35,  294,  2772,  28314,   306735, ...
%e A102539   6, 56,  672,  9504, 151008,  2617472, ...
%e A102539   7, 84, 1386, 28314, 674817, 18076916, ...
%e A102539   ...
%t A102539 T[n_, k_] := Product[(n + i + j - 1)/(i + j - 1), {i, 1, k}, {j, i, k}];
%t A102539 Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 06 2018 *)
%Y A102539 Rows include A000079, A001700, A003645, A000356.
%Y A102539 Main diagonal is A049505.
%K A102539 nonn,tabl
%O A102539 1,1
%A A102539 _Ralf Stephan_, Jan 14 2005