0,2

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

T. D. Noe, Table of n, a(n) for n=0..199

G.f. : (1-sqrt(1-4x))/(2x(1-x)) = C(x)/(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. a(n) = A014137[n+1]-1.

Cf. A002476.

Cf. A005554.

Cf. A137614.

Sequence in context: A018041 A073357 A047926 this_sequence A099324 A117420 A003101

Adjacent sequences: A014135 A014136 A014137 this_sequence A014139 A014140 A014141

nonn,nice

njas