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Square array read by antidiagonals arising in the enumeration of corners.
1

%I #21 Nov 07 2017 02:59:46

%S 1,2,2,5,16,5,14,91,91,14,42,456,936,456,42,132,2145,7425,7425,2145,

%T 132,429,9724,50765,85800,50765,9724,429,1430,43043,315315,805805,

%U 805805,315315,43043,1430,4862,187408,1831648,6584032,9962680,6584032,1831648,187408,4862,16796,806208,10127988,48674808,103698504,103698504,48674808,10127988,806208,16796

%N Square array read by antidiagonals arising in the enumeration of corners.

%C See Kreweras (1979) for precise definition.

%H G. Kreweras, <a href="http://dx.doi.org/10.1016/0012-365X(79)90163-8">Sur les extensions linéaires d'une famille particulière d'ordres partiels</a>, Discrete Math., 27 (1979), 279-295.

%H G. Kreweras, <a href="/A006330/a006330_1.pdf">Sur les extensions linéaires d'une famille particulière d'ordres partiels</a>, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)

%F Kreweras gives an explicit formula for the general term (see bottom display on page 291).

%e The first few antidiagonals are:

%e 1,

%e 2, 2,

%e 5, 16, 5,

%e 14, 91, 91, 14,

%e 42, 456, 936, 456, 42,

%e 132, 2145, 7425, 7425, 2145, 132,

%e ...

%t a[x_, y_] := (2(2x+2y+1)!(x^2+3x*y+y^2+4x+4y+3)) / (x!(x+1)!y!(y+1)!(x+y+1)(x+y+2)(x+y+3));

%t Table[Table[a[x-y, y], {y, 0, x}] // Reverse, {x, 0, 9}] // Flatten (* _Jean-François Alcover_, Aug 11 2017 *)

%Y The first row and column of the array are the Catalan numbers A000108.

%Y The second row and column are A214824.

%K nonn,tabl,easy

%O 0,2

%A _N. J. A. Sloane_, Jun 22 2015

%E More terms from _Jean-François Alcover_, Aug 11 2017