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

1,5

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]

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.

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)

Example

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;

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 *)

Crossrefs

Cf. A001006, A000108.

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

Sequence in context: A293472 A046726 A082137 * A318607 A340106 A259824

Adjacent sequences: A091184 A091185 A091186 * A091188 A091189 A091190

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 23 2004

Status

approved