A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)*C(n,s)*(C(n-s,s+1) - C(n-s-2,s-1)).

1, 1, 2, 1, 5, 1, 9, 5, 1, 14, 21, 1, 20, 56, 14, 1, 27, 120, 84, 1, 35, 225, 300, 42, 1, 44, 385, 825, 330, 1, 54, 616, 1925, 1485, 132, 1, 65, 936, 4004, 5005, 1287, 1, 77, 1365, 7644, 14014, 7007, 429

Offset: 3

Author

Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007

Keywords

• nonn
• tabf

Comments

Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.

Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012

Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. See also A108263 and A100754. - Tom Copeland, Oct 19 2014

Examples

			A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.
Triangle begins:
  1;
  1,   2;
  1,   5;
  1,   9,   5;
  1,  14,  21;
  1,  20,  56,  14;
  1,  27, 120,  84;
  1,  35, 225, 300,  42;
  1,  44, 385, 825, 330;
  ...
		

Links

• C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
• F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
• F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
• T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.

Crossrefs

Row sums are A007404.

Cf. A001263, A005043, A033282, A100754, A108263, A108767, A132081.

Programs

Magma

/* triangle excluding 0 */ [[Binomial(n,k)*Binomial(n-2-k,k)/(k+1): k in [0..n-3]]: n in [3..15]]; // Vincenzo Librandi, Oct 19 2014

Mathematica

Map[Most, Table[(1/n) Binomial[n, s] (Binomial[n - s, s + 1] - Binomial[n - s - 2, s - 1]), {n, 3, 14}, {s, 0, n}] /. k_ /; k <= 0 -> Nothing] // Flatten (* Michael De Vlieger, Jan 09 2016 *)

Formula

a(n,k) = binomial(n,k)*binomial(n-2-k,k)/(k+1). - David Callan, Jul 22 2008

From Peter Bala, Oct 22 2008: (Start)

O.g.f.: (1 + x - sqrt(1 - 2*x + x^2*(1 - 4*a)))/(2*x*(1 + a*x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . [corrected by Jason Yuen, Sep 22 2024]

Define a functional I on formal power series of the form f(x) = 1 + a*x + b*x^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).

Let now f(x) = 1 + a*x^2 + a*x^3 + a*x^4 + ... . Then the o.g.f. for this table is I(f(x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . Cf. A001263 and A108767. (End)

Extensions

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

Name corrected by Emeric Deutsch, Dec 20 2014