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

2, 3, 4, 4, 10, 8, 5, 20, 35, 16, 6, 35, 112, 126, 32, 7, 56, 294, 672, 462, 64, 8, 84, 672, 2772, 4224, 1716, 128, 9, 120, 1386, 9504, 28314, 27456, 6435, 256, 10, 165, 2640, 28314, 151008, 306735, 183040, 24310, 512, 11, 220, 4719, 75504, 674817

Offset: 1

Author

Ralf Stephan, Jan 14 2005

Keywords

• nonn
• tabl

Comments

Number of semistandard Young tableaux with at most n columns and with entries in [k].

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

Examples

			Square array T(n,k) begins:
  2,  4,    8,    16,     32,       64, ...
  3, 10,   35,   126,    462,     1716, ...
  4, 20,  112,   672,   4224,    27456, ...
  5, 35,  294,  2772,  28314,   306735, ...
  6, 56,  672,  9504, 151008,  2617472, ...
  7, 84, 1386, 28314, 674817, 18076916, ...
  ...
		

Links

• M. Lederer, A determinant-like formula for the Kostka numbers

Crossrefs

Rows include A000079, A001700, A003645, A000356.

Main diagonal is A049505.

Programs

Mathematica

T[n_, k_] := Product[(n + i + j - 1)/(i + j - 1), {i, 1, k}, {j, i, k}];
Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 06 2018 *)

Formula

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)].