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A078920 Upper triangle of Catalan Number Wall.

%I A078920 #95 Apr 05 2024 19:45:14
%S A078920 1,1,1,1,2,1,1,5,3,1,1,14,14,4,1,1,42,84,30,5,1,1,132,594,330,55,6,1,
%T A078920 1,429,4719,4719,1001,91,7,1,1,1430,40898,81796,26026,2548,140,8,1,1,
%U A078920 4862,379236,1643356,884884,111384,5712,204,9,1,1,16796,3711916,37119160,37119160,6852768,395352,11628,285,10,1
%N A078920 Upper triangle of Catalan Number Wall.
%C A078920 As square array: number of certain symmetric plane partitions, see Forrester/Gamburd paper.
%C A078920 Formatted as a square array, the column k gives the Hankel transform of the Catalan numbers (A000108) beginning at A000108(k); example: Hankel transform of [42, 132, 429, 1430, 4862, ...] is [42, 594, 4719, 26026, 111384, ...] (see A091962). - _Philippe Deléham_, Apr 12 2007
%C A078920 As square array T(n,k): number of all k-watermelons with a wall of length n. - _Ralf Stephan_, May 09 2007
%C A078920 Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used." de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." It turns out that for the current array, T(n,k) = b(n-k,k) for n >= 0 and 0 <= k <= n. - _Petros Hadjicostas_, Sep 04 2019
%C A078920 As square array, b(k, n) = T(n+k-1, n) for k >= 1 and n >= 1 is the number of n-tuples P = (p_1, p_2, ..., p_n) of non-intersecting lattice paths that lie below the diagonal, such that each p_i starts at (i, i) and ends at (2n+k-i, 2n+k-i). (This is just a different way of looking at n-watermelons with a wall of length k since many of the steps of these paths are going to be fixed while the rest form an n-watermelon. See the Krattenthaler et al. paper.) Equivalently b(k, n) is the number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths, each with 2k steps such that for every i (1 <= i <= n-1), p_i is included in p_{i+1}. A Dyck path p is said to be included in a Dyck path q if the height of path p after j steps is at most the height of path q after j steps, for all j (1 <= j <= 2k). - _Farzan Byramji_, Jun 17 2021
%H A078920 Alois P. Heinz, <a href="/A078920/b078920.txt">Rows n = 0..100, flattened</a>
%H A078920 R. Bacher, <a href="https://arxiv.org/abs/math/0109013">Matrices related to the Pascal triangle</a>, arXiv:math/0109013 [math.CO], 2001.
%H A078920 Paul Barry, <a href="https://arxiv.org/abs/2011.10827">Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers</a>, arXiv:2011.10827 [math.CO], 2020.
%H A078920 M. de Sainte-Catherine and G. Viennot, <a href="https://doi.org/10.1007/BFb0072509">Enumeration of certain Young tableaux with bounded height</a>, in: G. Labelle and P. Leroux (eds), <a href="https://doi.org/10.1007/BFb0072503">Combinatoire énumérative</a>, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
%H A078920 P. J. Forrester and A. Gamburd, <a href="https://arxiv.org/abs/math/0503002">Counting formulas associated with some random matrix averages</a>, arXiv:math/0503002 [math.CO], 2005.
%H A078920 P. J. Forrester and A. Gamburd, <a href="https://doi.org/10.1016/j.jcta.2005.09.001">Counting formulas associated with some random matrix averages</a>, J. Combin. Theory Ser. A 113(6) (2006), 934-951.
%H A078920 M. Fulmek, <a href="https://arxiv.org/abs/math/0607163">Asymptotics of the average height of 2-watermelons with a wall</a>, arXiv:math/0607163 [math.CO], 2006.
%H A078920 M. Fulmek, <a href="https://doi.org/10.37236/982">Asymptotics of the average height of 2-watermelons with a wall</a>, Electron. J. Combin. 14 (2007), R64.
%H A078920 C. Krattenthaler, A. J. Guttmann and X. G. Viennot, <a href="https://doi.org/10.1088/0305-4470/33/48/318">Vicious walkers, friendly walkers and Young tableaux: II. With a wall</a>, J. Phys. A: Math. Gen. 33 (2000), 8835-8866.
%H A078920 Vincent Pilaud, <a href="http://arxiv.org/abs/1505.07665">Brick polytopes, lattice quotients, and Hopf algebras</a>, arXiv:1505.07665 [math.CO], 2015.
%H A078920 Vincent Pilaud, <a href="https://doi.org/10.1016/j.jcta.2017.11.014">Brick polytopes, lattice quotients, and Hopf algebras</a>, J. Combin. Theory Ser. A 155 (2018), 418-457.
%H A078920 Michael Somos, <a href="https://grail.eecs.csuohio.edu/~somos/nwic.html">Number Walls in Combinatorics</a>.
%F A078920 T(n,k) = Product_{i=1..n-k} Product_{j=i..n-k} (i+j+2*k)/(i+j). [corrected by _Petros Hadjicostas_, Jul 24 2019]
%F A078920 From _G. C. Greubel_, Dec 17 2021: (Start)
%F A078920 T(n, k) = Product_{j=0..k-1} binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j).
%F A078920 T(n, k) = ((n+1)!/(n-k+1)!)*Product_{j=0..k-1} Catalan(n-j)/binomial(n+j+1, n-j). (End)
%e A078920 Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
%e A078920   1;
%e A078920   1,    1;
%e A078920   1,    2,      1;
%e A078920   1,    5,      3,       1;
%e A078920   1,   14,     14,       4,      1;
%e A078920   1,   42,     84,      30,      5,      1;
%e A078920   1,  132,    594,     330,     55,      6,    1;
%e A078920   1,  429,   4719,    4719,   1001,     91,    7,   1;
%e A078920   1, 1430,  40898,   81796,  26026,   2548,  140,   8, 1;
%e A078920   1, 4862, 379236, 1643356, 884884, 111384, 5712, 204, 9, 1;
%e A078920   ...
%p A078920 T:= (n, k)-> mul(mul((i+j+2*k)/(i+j), j=i..n-k), i=1..n-k):
%p A078920 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Sep 04 2019
%t A078920 T[n_, k_] := Product[(2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!, {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 28 2015, adapted from PARI *)
%o A078920 (PARI) T(n,k)=if(k<0 || k>n,0,prod(i=0,k-1,(2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!))
%o A078920 (PARI) {C(n)=if(n<0,0,(2*n)!/n!/(n+1)!)}; T(n,k)=if(k<0 || k>n,0,matdet(matrix(k,k,i,j,C(i+j-1+n-k))))
%o A078920 (Sage)
%o A078920 def A078920(n,k): return product( binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j) for j in (0..k-1) )
%o A078920 flattened([[A078920(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Dec 17 2021
%Y A078920 Columns are A000012, A000108, A005700, A006149, A006150, A006151.
%Y A078920 Diagonals are A000027, A000330, A006858.
%Y A078920 T(2n,n) gives A358597.
%Y A078920 Cf. A123352.
%K A078920 easy,nonn,tabl
%O A078920 0,5
%A A078920 _Michael Somos_, Dec 15 2002
%E A078920 T(0,0) = 1 prepended by _Petros Hadjicostas_, Jul 24 2019