A102539 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..50.

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

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)].

EXAMPLE

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, ...

...

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 *)

CROSSREFS

Rows include A000079, A001700, A003645, A000356.

Main diagonal is A049505.

Sequence in context: A217478 A279788 A207627 * A240220 A250229 A250277

Adjacent sequences: A102536 A102537 A102538 * A102540 A102541 A102542

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Jan 14 2005

STATUS

approved