A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

1, 2, 1, 5, 6, 2, 14, 28, 20, 5, 42, 120, 135, 70, 14, 132, 495, 770, 616, 252, 42, 429, 2002, 4004, 4368, 2730, 924, 132, 1430, 8008, 19656, 27300, 23100, 11880, 3432, 429, 4862, 31824, 92820, 157080, 168300, 116688, 51051, 12870, 1430

Offset: 0

Author

N. J. A. Sloane, Jan 11 2014

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1,
     2,    1,
     5,    6,     2,
    14,   28,    20,     5,
    42,  120,   135,    70,    14,
   132,  495,   770,   616,   252,    42,
   429, 2002,  4004,  4368,  2730,   924,  132,
  1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
  ...
		

Links

• Reinhard Zumkeller, Rows n=0..125 of triangle, flattened
• Antoine Abram, Florent Hivert, James D. Mitchell, Jean-Christophe Novelli, and Maria Tsalakou, Power Quotients of Plactic-like Monoids, arXiv:2406.16387 [math.CO], 2024. See p. 5.
• Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
• Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
• Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
• Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
• G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
• C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.
• Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
• A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
• Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2. See pp. 21-22. - _N. J. A. Sloane_, Jul 12 2014

Crossrefs

A062991 is a signed version. See also A094385 for another version.

Cf. A009766.

The two borders give the Catalan numbers A000108.

Cf. A062992 (row sums).

The second and third columns give A002694 and A244887.

Programs

Haskell

a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
a234950_row n = map (a234950 n) [0..n]
a234950_tabl = map a234950_row [0..]
-- Reinhard Zumkeller, Jan 12 2014

Maple

T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Sep 04 2018

Mathematica

T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

PARI

T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Sep 06 2015

Formula

G.f.: 1/x*(1-sqrt(1-4*x-4*x*y))/(1+2*y+sqrt(1-4*x-4*x*y)). - Vladimir Kruchinin, Sep 04 2018

T(n,k) = 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)). - Peter Luschny, Sep 04 2018