cp's OEIS Frontend

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.

%I A234950 #67 Jun 28 2024 14:55:14
%S A234950 1,2,1,5,6,2,14,28,20,5,42,120,135,70,14,132,495,770,616,252,42,429,
%T A234950 2002,4004,4368,2730,924,132,1430,8008,19656,27300,23100,11880,3432,
%U A234950 429,4862,31824,92820,157080,168300,116688,51051,12870,1430
%N 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.
%H A234950 Reinhard Zumkeller, <a href="/A234950/b234950.txt">Rows n=0..125 of triangle, flattened</a>
%H A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 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 A234950 Triangle begins:
%e A234950      1,
%e A234950      2,    1,
%e A234950      5,    6,     2,
%e A234950     14,   28,    20,     5,
%e A234950     42,  120,   135,    70,    14,
%e A234950    132,  495,   770,   616,   252,    42,
%e A234950    429, 2002,  4004,  4368,  2730,   924,  132,
%e A234950   1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
%e A234950   ...
%p A234950 T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):
%p A234950 seq(seq(T(n,k), k=0..n), n=0..8); # _Peter Luschny_, Sep 04 2018
%t A234950 T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
%t A234950 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018, from Maple *)
%o A234950 (Haskell)
%o A234950 a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
%o A234950 a234950_row n = map (a234950 n) [0..n]
%o A234950 a234950_tabl = map a234950_row [0..]
%o A234950 -- _Reinhard Zumkeller_, Jan 12 2014
%o A234950 (PARI) T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
%o A234950 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ _Michel Marcus_, Sep 06 2015
%Y A234950 A062991 is a signed version. See also A094385 for another version.
%Y A234950 Cf. A009766.
%Y A234950 The two borders give the Catalan numbers A000108.
%Y A234950 Cf. A062992 (row sums).
%Y A234950 The second and third columns give A002694 and A244887.
%K A234950 nonn,tabl
%O A234950 0,2
%A A234950 _N. J. A. Sloane_, Jan 11 2014