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

0,6

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

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.

Example

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;
...

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 *)

Crossrefs

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

Sequence in context: A046766 A292147 A003285 * A071496 A071502 A074704

Adjacent sequences: A059344 A059345 A059346 * A059348 A059349 A059350

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Jan 27 2001

Extensions

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

Status

approved