A072638 Number of unary-binary rooted trees of height at most n.

0, 1, 3, 10, 66, 2278, 2598060, 3374961778891, 5695183504492614029263278, 16217557574922386301420536972254869595782763547560

Offset: 0

Author

Antti Karttunen, Jun 02 2002

Keywords

• nonn

Comments

A unary-binary tree is one in which the degree of every node is <= 3.

a(n+1) = (a(n)+1)-th triangular numbers = A000217(a(n)+1). a(n+1) = (a(n) + 1) * (a(n) + 2) / 2. a(n+1) = A006894(n+2) - 1. - Jaroslav Krizek, Sep 11 2009

a(n) is the smallest integer that is the sum of n distinct members of the complete sequence A000124. See A204009 for the binary vectors that select the terms from A000124. - Frank M Jackson, Jan 09 2012

Links

• Index entries for sequences related to rooted trees

Crossrefs

Maximal position in A071673 where the value n occurs.

Binary width of each term: A072641. Cf. A072639, A072640, A072654.

Programs

Maple

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n], n=-1..9); # Zerinvary Lajos, Jun 08 2007

Mathematica

Clear[a]; a[0] = 0; a[n_] := a[n] = 1 + (a[n-1]*(a[n-1]+3))/2; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 31 2013 *)

Formula

a(n+1) = 1 + (a(n)*(a(n)+3))/2.

Conjecture: a(n) = A006894(n+1) - 1. - R. J. Mathar, Apr 23 2007

a(n) := C(a(n-1) + 2, 2), n >= -1. - Zerinvary Lajos, Jun 08 2007

Extensions

Edited by Christian G. Bower, Oct 23 2002