0,3

Number of paths starting from the root in all ordered trees with n+1 edges (a path is a nonempty tree with no vertices of outdegree greater than 1). Example: a(2)=8 because the five trees with three edges have altogether 1+0+2+2+3=8 paths hanging from the roots. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 20 2002

a(n)=sum of the mean maximal pyramid size over all Dyck (n+1)-paths. Also, a(n)=sum of the mean maximal sawtooth size over all Dyck (n+1)-paths. A pyramid (resp. sawtooth) in a Dyck path is a subpath of the form U^k D^k (resp. (UD)^k) with k>=1 and k is its size. For example, the maximal pyramids in the Dyck path uUUDD|UD|UDdUUDD are indicated by uppercase letters (and separated by a vertical bar). Their sizes are 2,1,1,2 left to right and the mean maximal pyramid size of the path is 6/4=3/2. Also, the mean maximal sawtooth size of this path is (1+2+1)/3=4/3. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

p^2 divides a(p-1) for prime p of form p=6k+1 (A002476(k)). - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006

p^2 divides a(p^2-1) for prime p>3. p^2 divides a(p^3-1) for prime p=7,13,19.. prime p in the form p=6k+1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006

Row sums of triangle A137614 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 30 2008

Equals INVERTi transform of A095930: (1, 4, 15, 57, 220, 859,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

T. D. Noe, Table of n, a(n) for n=0..200

a(n) = A014137(n)-1.

G.f.: (1-2*x-sqrt(1-4x))/(2x(1-x)) = (C(x)-1)/(1-x) where C(x) is the generating function for the Catalan numbers. - Rocio Blanco, Apr 02 2007

a(n) = Sum[ CatalanNumber[k], {k,1,n}]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006

Binomial transform of A005554: (1, 2, 3, 6, 13, 30, 72,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007

a:=n->sum((binomial(2*j, j)/(j+1)), j=1..n): seq(a(n), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

Table[Sum[(2k)!/k!/(k+1)!, {k, 1, n}], {n, 1, 70}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006

Cf. A000108, A002476, A005554, A137614, A095930.

Sequence in context: A073357 A164934 A047926 this_sequence A099324 A117420 A003101

Adjacent sequences: A014135 A014136 A014137 this_sequence A014139 A014140 A014141

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Edited by Max Alekseyev, Sep 13 2009 (including adding an initial 0)

Definition edited by N. J. A. Sloane, Oct 03 2009