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

0,2

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

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

Example

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,
  ...

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 *)

Prog

(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
(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

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.

Sequence in context: A185384 A274728 A062991 * A275228 A118984 A073474

Adjacent sequences: A234947 A234948 A234949 * A234951 A234952 A234953

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 11 2014

Status

approved