A059365 Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1) - binomial(2*r-s-1,r), r>=0, 0 <= s <= r.

0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 14, 14, 9, 4, 1, 0, 42, 42, 28, 14, 5, 1, 0, 132, 132, 90, 48, 20, 6, 1, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44

Offset

0,8

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.

D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.

FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path., The number of initial rises of a Dyck paths., The number of nodes on the left branch of the tree., The number of subtrees.

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.

Index to sequences related to Catalan

Formula

Essentially the same triangle as [0, 1, 1, 1, 1, 1, 1, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938, but the first term is T(0,0) = 0.

Example

Triangle starts
  0;
  0,    1;
  0,    1,    1;
  0,    2,    2,    1;
  0,    5,    5,    3,    1;
  0,   14,   14,    9,    4,    1;
  0,   42,   42,   28,   14,    5,   1;
  0,  132,  132,   90,   48,   20,   6,   1;
  0,  429,  429,  297,  165,   75,  27,   7,  1;
  0, 1430, 1430, 1001,  572,  275, 110,  35,  8, 1;
  0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1;
  ...

Mathematica

Table[Binomial[2*r - s - 1, r - 1] - Binomial[2*r - s - 1, r], {r, 0, 10}, {s, 0, r}] // Flatten (* G. C. Greubel, Jan 08 2017 *)

Prog

(PARI) tabl(nn) = { print(0); for (r=1, nn, for (s=0, r, print1(binomial(2*r-s-1, r-1)-binomial(2*r-s-1, r), ", "); ); print(); ); }  \\ Michel Marcus, Nov 01 2013
(Magma) /* as triangle */ [[[0] cat [Binomial(2*r-s-1, r-1)- Binomial(2*r-s-1, r): s in [1..r]]: r in [0..10]]]; // Vincenzo Librandi, Jan 09 2017

Crossrefs

See also the triangle in A009766. First 2 diagonals both give A000108, next give A000245, A002057.

Cf. A009766, A000007, A084938, A000108.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.

Essentially the same as A033184.

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A053121, A059365, A099039, A106566, A130020, A047072, A171567, A181645.

Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...

Sequence in context: A128497 A011434 A147746 * A106566 A099039 A205574

Adjacent sequences: A059362 A059363 A059364 * A059366 A059367 A059368

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 28 2001

Status

approved