A155520 Triangle read by rows: A(n,k) is the number of ordered trees with n edges having k drawings.

1, 2, 3, 2, 4, 2, 6, 1, 1, 5, 2, 6, 9, 1, 4, 4, 4, 2, 1, 2, 2

Offset

1,2

Comments

A drawing of an ordered tree T with n edges is a sequence of trees (T_0, T_1, T_2, ..., T_n), such that T_n = T and T_{i-1} arises from T_i by deleting a leaf of T_i.

Row sums are the Catalan numbers (A000108).

Links

Table of n, a(n) for n=1..21.

Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

Formula

Sum_{k>0} k*A(n,k) = A014307(n).

Example

We represent ordered trees by their corresponding Dyck paths via the "glove" bijection.
The "tree" UDUUDD has 2 drawings: *, UD, UUDD, UDUUDD and *, UD, UDUD, UDUUDD;
the "tree" UUDDUD has 2 drawings: *, UD, UUDD, UUDDUD and *, UD, UUDD, UUDDUD.
Thus A(3,2)=2.
The "tree" UUUDDD has 1 drawing: *, UD, UUDD, UUUDDD;
the "tree" UUDUDD has 1 drawing: *, UD, UUDD, UUDUDD;
the "tree" UDUDUD has 1 drawing: *, UD, UDUD, UDUDUD.
Thus A(3,1)=3.
Triangle starts:
  1;
  2;
  3, 2;
  4, 2, 6, 1, 1;
  5, 2, 6, 9, 1, 4, 4, 4, 2, 1, 2, 2;
  ...

Crossrefs

Cf. A000108 (row sums), A014307.

Sequence in context: A322587 A372569 A058973 * A365784 A235912 A339749

Adjacent sequences: A155517 A155518 A155519 * A155521 A155522 A155523

Keyword

nonn,tabf,more

Author

Emeric Deutsch, Mar 19 2009

Extensions

Keyword tabf added by Michel Marcus, Apr 09 2013

Status

approved