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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Steve Butler, R. Graham, C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.

Yue Cai, Catherine Yan, Counting with Borel's triangle, Texas A&M University.

Yue Cai, Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.

G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.

C. A. Francisco, J. Mermin, J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.

A. Lakshminarayan, Z. Puchala, 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

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,

  ...

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

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Last modified May 27 13:15 EDT 2018. Contains 304690 sequences. (Running on oeis4.)