A181645 -id:A181645 - cp's OEIS frontend

A059365 Another version of the Catalan triangle: T(r,s) = binomial(2r-s-1,r-1) - binomial(2r-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

N. J. A. Sloane, Jan 28 2001

			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;
  ...

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

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, ...

/* 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

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 *)

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

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.

A181644 Eigentriangle for the Catalan triangle (c(x), xc(x)), A033184.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 22, 11, 4, 1, 1, 92, 46, 17, 5, 1, 1, 426, 213, 79, 24, 6, 1, 1, 2150, 1075, 399, 122, 32, 7, 1, 1, 11708, 5854, 2173, 665, 176, 41, 8, 1, 1, 68282, 34141, 12673, 3878, 1027, 242, 51, 9, 1, 1, 423948, 211974, 78683, 24075, 6373, 1502

Offset: 0

Paul Barry, Nov 03 2010

First column is essentially A091768. Inverse of A181645.

			Triangle begins
  1,
  1, 1,
  2, 1, 1,
  6, 3, 1, 1,
  22, 11, 4, 1, 1,
  92, 46, 17, 5, 1, 1,
  426, 213, 79, 24, 6, 1, 1,
  2150, 1075, 399, 122, 32, 7, 1, 1,
  11708, 5854, 2173, 665, 176, 41, 8, 1, 1

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A033184, A091768, A181645.

cp's OEIS Frontend

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

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A181644 Eigentriangle for the Catalan triangle (c(x), xc(x)), A033184.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A181644 Eigentriangle for the Catalan triangle (c(x), xc(x)), A033184.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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