A091187 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 9, 1, 5, 20, 40, 45, 21, 1, 6, 30, 80, 135, 126, 51, 1, 7, 42, 140, 315, 441, 357, 127, 1, 8, 56, 224, 630, 1176, 1428, 1016, 323, 1, 9, 72, 336, 1134, 2646, 4284, 4572, 2907, 835, 1, 10, 90, 480, 1890, 5292, 10710, 15240, 14535, 8350, 2188

Offset: 1

Author

Emeric Deutsch, Feb 23 2004

Keywords

• nonn
• tabl

Comments

Row sums are the Catalan numbers A000108. Diagonal entries are the Motzkin numbers A001006.

Equals binomial transform of an infinite lower triangular matrix with A001006 as the main diagonal and the rest zeros. [Gary W. Adamson, Dec 31 2008] [Corrected by Paul Barry, Mar 06 2011]

Reversal of A091869. Diagonal sums are A026418(n+2). [Paul Barry, Mar 06 2011]

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   4;
  1, 4, 12,  16,   9;
  1, 5, 20,  40,  45,  21;
  1, 6, 30,  80, 135, 126,  51;
  1, 7, 42, 140, 315, 441, 357, 127;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
• J.-L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
• J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
• Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Crossrefs

Cf. A001006, A000108.

Cf. A007476. [Gary W. Adamson, Dec 31 2008]

Programs

Maple

M := n->sum(binomial(n+1,q)*binomial(n+1-q,q-1),q=0..ceil((n+1)/2))/(n+1): T := (n,k)->binomial(n-1,k-1)*M(k-1): seq(seq(T(n,k),k=1..n),n=1..13);

Mathematica

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, k_] := m[k - 1]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)

Formula

T(n,k) = M(k-1)*binomial(n-1, k-1), where M(k) = A001006(k) = (Sum_{q=0..ceiling((k+1)/2)} binomial(k+1, q)*binomial(k+1-q, q-1))/(k+1) is a Motzkin number.

G.f.: G = G(t,z) satisfies t*z*G^2 -(1 - z + t*z)*G + 1- z + t*z = 0.

From Paul Barry, Mar 06 2011: (Start)

G.f.: 1/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-... (continued fraction).

G.f.: (1-x(1+y)-sqrt(1-2x(1+y)+x^2(1+2y-3y^2)))/(2x^2*y^2).

E.g.f.: exp(x(1+y))*Bessel_I(1,2*x*y)/(x*y). (End)