%I #60 Jun 25 2023 09:38:18
%S 1,1,3,1,8,12,1,15,55,55,1,24,156,364,273,1,35,350,1400,2380,1428,1,
%T 48,680,4080,11628,15504,7752,1,63,1197,9975,41895,92169,100947,43263,
%U 1,80,1960,21560,123970,396704,708400,657800,246675,1,99,3036,42504
%N Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.
%C Number of dissections of a convex (2n+2)-gon by k-1 noncrossing diagonals into (2j+2)-gons, 1 <= j <= n-1.
%C Apparently, a signed, refined version of this array is given on page 65 of the Einziger link, related to the antipode of a Hopf algebra. - _Tom Copeland_, May 19 2015
%C The f-vectors of the simplicial noncrossing hypertree complexes of McCammond (p. 15). The reduced Euler characteristics are the signed Catalan numbers A000108. - _Tom Copeland_, May 19 2017
%C The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*(n)!) in the basis made of the binomial(x+i,i). - _F. Chapoton_, Nov 01 2022
%C Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*n!) = Sum_{k = 1..n} (-1)^(k+1)*T(n,n+1-k)*binomial(x+3*n+1-k, 3*n+1-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6)*(x+7)*(x+5)(x+6)/(7!*3!) = 12*binomial(x+9,9) - 8*binomial(x+8,8) + binomial(x+7,7). - _Peter Bala_, Jun 25 2023
%H Michael De Vlieger, <a href="/A102537/b102537.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150).
%H H. Einziger, <a href="https://search.proquest.com/openview/9a6007300d492143210f01408f4f0e70/1?pq-origsite=gscholar&cbl=18750&diss=y">Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions</a>, Dissertation (2010), George Washington University.
%H J. McCammond, <a href="http://web.math.ucsb.edu/~jon.mccammond/papers/index.html">Noncrossing Hypertrees</a>, 2015.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://arxiv.org/abs/2106.08257">Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra</a>, arXiv:2106.08257 [math.CO], 2021-2022.
%H E. Tzanaki, <a href="https://arxiv.org/abs/math/0501100">Polygon dissections and some generalizations of cluster complexes</a>, arXiv:math/0501100 [math.CO], 2005.
%e Triangle begins
%e 1;
%e 1, 3;
%e 1, 8, 12;
%e 1, 15, 55, 55;
%e 1, 24, 156, 364, 273;
%e 1, 35, 350, 1400, 2380, 1428;
%e 1, 48, 680, 4080, 11628, 15504, 7752;
%e 1, 63, 1197, 9975, 41895, 92169, 100947, 43263;
%e 1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675;
%t Table[1/n*Binomial[2 n + k, k - 1] Binomial[n, k], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, May 20 2017 *)
%o (Magma) [[1/n * Binomial(2*n+k,k-1) * Binomial(n,k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, May 20 2015
%Y Left-hand columns include A005563. Right-hand columns include essentially A001764 and A013698.
%Y Row sums are in A003168.
%Y Cf. A000108, A120986.
%Y Cf. A243662 for rows reversed.
%K nonn,tabl
%O 1,3
%A _Ralf Stephan_, Jan 14 2005