A234950 - OEIS

%I #67 Jun 28 2024 14:55:14

%S 1,2,1,5,6,2,14,28,20,5,42,120,135,70,14,132,495,770,616,252,42,429,

%T 2002,4004,4368,2730,924,132,1430,8008,19656,27300,23100,11880,3432,

%U 429,4862,31824,92820,157080,168300,116688,51051,12870,1430

%N 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.

%H Reinhard Zumkeller, <a href="/A234950/b234950.txt">Rows n=0..125 of triangle, flattened</a>

%H Antoine Abram, Florent Hivert, James D. Mitchell, Jean-Christophe Novelli, and Maria Tsalakou, <a href="https://arxiv.org/abs/2406.16387">Power Quotients of Plactic-like Monoids</a>, arXiv:2406.16387 [math.CO], 2024. See p. 5.

%H Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.

%H Steve Butler, R. Graham, and C. H. Yan, <a href="http://www.math.ucsd.edu/~ronspubs/17_03_parking.pdf">Parking distributions on trees</a>, European Journal of Combinatorics 65 (2017), 168-185.

%H Yue Cai and Catherine Yan, <a href="http://www.math.tamu.edu/~catherine.yan/Files/Borel-main-final.pdf">Counting with Borel's triangle</a>, Texas A&M University.

%H Yue Cai and Catherine Yan, <a href="https://arxiv.org/abs/1804.01597">Counting with Borel's triangle</a>, arXiv:1804.01597 [math.CO], 2018.

%H G. Chatel and V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">Cambrian Hopf Algebras</a>, arXiv:1411.3704 [math.CO], 2014-2015.

%H C. A. Francisco, J. Mermin, and J. Schweig, <a href="http://www.math.okstate.edu/~jayjs/ppt.pdf">Catalan numbers, binary trees, and pointed pseudotriangulations</a>, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.

%H Lord C. Kavi and Michael W. Newman, <a href="https://arxiv.org/abs/2212.08795">Counting closed walks in infinite regular trees using Catalan and Borel's triangles</a>, arXiv:2212.08795 [math.CO], 2022.

%H A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, <a href="http://arxiv.org/abs/1407.1169">Diagonal unitary entangling gates and contradiagonal quantum states</a>, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.

%H Jeffrey B. Remmel, <a href="https://doi.org/10.37236/3210">Consecutive Up-down Patterns in Up-down Permutations</a>, Electron. J. Combin., 21 (2014), #P3.2. See pp. 21-22. - _N. J. A. Sloane_, Jul 12 2014

%F 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

%F 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

%e Triangle begins:

%e      1,

%e      2,    1,

%e      5,    6,     2,

%e     14,   28,    20,     5,

%e     42,  120,   135,    70,    14,

%e    132,  495,   770,   616,   252,    42,

%e    429, 2002,  4004,  4368,  2730,   924,  132,

%e   1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,

%e   ...

%p T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):

%p seq(seq(T(n,k), k=0..n), n=0..8); # _Peter Luschny_, Sep 04 2018

%t T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018, from Maple *)

%o (Haskell)

%o a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]

%o a234950_row n = map (a234950 n) [0..n]

%o a234950_tabl = map a234950_row [0..]

%o -- _Reinhard Zumkeller_, Jan 12 2014

%o (PARI) T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ _Michel Marcus_, Sep 06 2015

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

%Y Cf. A009766.

%Y The two borders give the Catalan numbers A000108.

%Y Cf. A062992 (row sums).

%Y The second and third columns give A002694 and A244887.

%K nonn,tabl

%O 0,2

%A _N. J. A. Sloane_, Jan 11 2014