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A014138
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Partial sums of (Catalan numbers starting 1,2,5,...), cf. A000108.
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271
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0, 1, 3, 8, 22, 64, 196, 625, 2055, 6917, 23713, 82499, 290511, 1033411, 3707851, 13402696, 48760366, 178405156, 656043856, 2423307046, 8987427466, 33453694486, 124936258126, 467995871776, 1757900019100
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OFFSET
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0,3
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COMMENTS
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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, 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, Jun 07 2006
p^2 divides a(p-1) for prime p of form p=6k+1 (A002476(k)). - Alexander Adamchuk, 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, Jul 03 2006
Row sums of triangle A137614 - Gary W. Adamson, Jan 30 2008
Equals INVERTi transform of A095930: (1, 4, 15, 57, 220, 859,...). [From Gary W. Adamson, May 15 2009]
a(n) < A000108(n+1), therefore A176137(n) <= 1. [From Reinhard Zumkeller, Apr 10 2010]
a(n) is also the sum of the numbers in Catalan's triangle (A009766)from row 0 to row n. [From Patrick Labarque (patrick.labarque(AT)base.be), Jul 27 2010]
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, Arxiv preprint arXiv:1107.5490, 2011.
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FORMULA
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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, Jul 03 2006
Binomial transform of A005554: (1, 2, 3, 6, 13, 30, 72,...). - Gary W. Adamson, Nov 23 2007
Conjecture: (n+1)*a(n) +(1-5n)*a(n-2) +2*(2n-1)*a(n-2)=0. - R. J. Mathar, Dec 14 2011
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MAPLE
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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
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MATHEMATICA
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Table[Sum[(2k)!/k!/(k+1)!, {k, 1, n}], {n, 1, 70}] - Alexander Adamchuk, Jul 03 2006
Join[{0}, Accumulate[CatalanNumber[Range[30]]]] (* Harvey P. Dale, Jan 25 2013 *)
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PROG
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(PARI) Vec((1-2*x-(1-4x)^(1/2))/(2x(1-x))) \\ Charles R Greathouse IV, Feb 11 2011
(Haskell)
a014138 n = a014138_list !! n
a014138_list = scanl1 (+) a000108_list -- Reinhard Zumkeller, Mar 01 2013
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CROSSREFS
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Cf. A000108, A002476, A005554, A137614, A095930.
Cf. A155587.
Sequence in context: A164934 A047926 A192681 * A099324 A117420 A003101
Adjacent sequences: A014135 A014136 A014137 * A014139 A014140 A014141
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KEYWORD
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nonn,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by Max Alekseyev, Sep 13 2009 (including adding an initial 0)
Definition edited by N. J. A. Sloane, Oct 03 2009
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STATUS
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approved
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