A102537 Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.

1, 1, 3, 1, 8, 12, 1, 15, 55, 55, 1, 24, 156, 364, 273, 1, 35, 350, 1400, 2380, 1428, 1, 48, 680, 4080, 11628, 15504, 7752, 1, 63, 1197, 9975, 41895, 92169, 100947, 43263, 1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675, 1, 99, 3036, 42504

Offset: 1

Author

Ralf Stephan, Jan 14 2005

Keywords

• nonn
• tabl

Comments

Number of dissections of a convex (2n+2)-gon by k-1 noncrossing diagonals into (2j+2)-gons, 1 <= j <= n-1.

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

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

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

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

Examples

			Triangle begins
  1;
  1,  3;
  1,  8,   12;
  1, 15,   55,    55;
  1, 24,  156,   364,    273;
  1, 35,  350,  1400,   2380,   1428;
  1, 48,  680,  4080,  11628,  15504,   7752;
  1, 63, 1197,  9975,  41895,  92169, 100947,  43263;
  1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
• H. Einziger, Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions, Dissertation (2010), George Washington University.
• J. McCammond, Noncrossing Hypertrees, 2015.
• Jean-Christophe Novelli and Jean-Yves Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
• Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra, arXiv:2106.08257 [math.CO], 2021-2022.
• E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, arXiv:math/0501100 [math.CO], 2005.

Crossrefs

Left-hand columns include A005563. Right-hand columns include essentially A001764 and A013698.

Row sums are in A003168.

Cf. A000108, A120986.

Cf. A243662 for rows reversed.

Programs

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

Mathematica

Table[1/n*Binomial[2 n + k, k - 1] Binomial[n, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)