A059347 Difference array of Motzkin numbers A001006 read by antidiagonals.

1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 2, 2, 3, 5, 9, 0, 2, 4, 7, 12, 21, 5, 5, 7, 11, 18, 30, 51, 0, 5, 10, 17, 28, 46, 76, 127, 14, 14, 19, 29, 46, 74, 120, 196, 323, 0, 14, 28, 47, 76, 122, 196, 316, 512, 835, 42, 42, 56, 84, 131, 207, 329, 525, 841, 1353, 2188, 0, 42, 84, 140, 224

Offset: 0

Author

N. J. A. Sloane, Jan 27 2001

Keywords

• nonn
• easy
• nice
• tabl

Comments

Row sums of odd rows (e.g., 4 = 1+1+2 for 3rd row) equal the Motzkin number of next row. Row sums of even rows equal the Motzkin number of the next row - n!/((n/2)!((n/2)+1)!) (i.e., A001006(n) - A000108(n/2) where A000108 are the Catalan numbers). - Gerald McGarvey, Dec 05 2004

Examples

			Triangle begins:
1;
0,1;
1,1,2;
0,1,2,4;
2,2,3,5,9;
0,2,4,7,12,21;
5,5,7,11,18,30,51;
...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Crossrefs

Top row is A001006, leading diagonals give A000108 (interspersed with 0's), A000108 doubled up, A059348.

Programs

Mathematica

max = 12; A001006 = CoefficientList[ Series[ (1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, max}], x] ; row[0] = A001006; row[n_] := Differences[A001006, n]; Flatten[ Table[ row[n-k][[k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 12 2012, from formula *)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001