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

0, 1, 3, 10, 66, 2278, 2598060, 3374961778891, 5695183504492614029263278, 16217557574922386301420536972254869595782763547560, 131504586847961235687181874578063117114329409897615188504091716162522225834932122128288032336298141

Offset

0,3

Comments

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

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

Table of n, a(n) for n=0..10.

Index entries for sequences related to rooted trees

Formula

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

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

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

From Jaroslav Krizek, Sep 11 2009: (Start)

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. (End)

Maple

a[0]:=0: for n to 10 do a[n]:= binomial(a[n-1]+2, 2) od: seq(a[n], n=0..10); # 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 *)

Prog

(PARI) first(nn)= my(t=-1); vector(nn, n, t=t*(t+3)/2+1) \\ Ruud H.G. van Tol, Feb 07 2026

Crossrefs

Maximal position in A071673 where the value n occurs.

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

Cf. A006894.

Sequence in context: A217388 A390165 A004102 * A262843 A080526 A393074

Adjacent sequences: A072635 A072636 A072637 * A072639 A072640 A072641

Keyword

nonn,easy

Author

Antti Karttunen, Jun 02 2002

Extensions

Edited by Christian G. Bower, Oct 23 2002

a(10) added by Ruud H.G. van Tol, Feb 08 2026

Status

approved