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 A014138 Partial sums of (Catalan numbers starting 1, 2, 5, ...). 288

%I

%S 0,1,3,8,22,64,196,625,2055,6917,23713,82499,290511,1033411,3707851,

%T 13402696,48760366,178405156,656043856,2423307046,8987427466,

%U 33453694486,124936258126,467995871776,1757900019100

%N Partial sums of (Catalan numbers starting 1, 2, 5, ...).

%C 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

%C a(n) is the 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

%C p^2 divides a(p-1) for prime p of form p=6k+1 (A002476(k)). - _Alexander Adamchuk_, Jul 03 2006

%C 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

%C Row sums of triangle A137614. - _Gary W. Adamson_, Jan 30 2008

%C Equals INVERTi transform of A095930: (1, 4, 15, 57, 220, 859, ...). - _Gary W. Adamson_, May 15 2009

%C a(n) < A000108(n+1), therefore A176137(n) <= 1. - _Reinhard Zumkeller_, Apr 10 2010

%C a(n) is also the sum of the numbers in Catalan's triangle (A009766) from row 0 to row n. - _Patrick Labarque_, Jul 27 2010

%C Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - _Gary W. Adamson_, May 20 2013

%C p divides a((p-3)/2) for primes {11,23,47,59,...} = A068231 primes congruent to 11 mod 12. - _Alexander Adamchuk_, Dec 27 2013

%H G. C. Greubel, <a href="/A014138/b014138.txt">Table of n, a(n) for n = 0..1000</a>(terms 0 to 200 computed by T. D. Noe)

%H P. Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

%H S. B. Ekhad, M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt"> Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)

%H Kevin Topley, <a href="http://arxiv.org/abs/1601.04223">Computationally Efficient Bounds for the Sum of Catalan Numbers</a>, arXiv:1601.04223 [math.CO], 2016.

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

%F 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

%F a(n) = Sum_{k=1..n} A000108(k). - _Alexander Adamchuk_, Jul 03 2006

%F Binomial transform of A005554: (1, 2, 3, 6, 13, 30, 72, ...). - _Gary W. Adamson_, Nov 23 2007

%F Conjecture: (n+1)*a(n) + (1-5n)*a(n-1) + 2*(2n-1)*a(n-2) = 0. - _R. J. Mathar_, Dec 14 2011

%F Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - _Gary W. Adamson_, May 20 2013

%F G.f.: 1/x - G(0)/(1-x)/x, where G(k)= 1 - x/(1 - x/(1 - x/(1 - x/G(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013

%F G.f.: 1/x - T(0)/(2*x*(1-x)), where T(k) = 2*x*(2*k+1)+ k+2 - 2*x*(k+2)*(2*k+3)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2013

%F a(n) ~ 2^(2*n+2)/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Dec 10 2013

%p a:=n->sum((binomial(2*j,j)/(j+1)),j=1..n): seq(a(n), n=0..24); # _Zerinvary Lajos_, Dec 01 2006

%t Table[Sum[(2k)!/k!/(k+1)!,{k,1,n}],{n,1,70}] (* _Alexander Adamchuk_, Jul 03 2006 *)

%t Join[{0},Accumulate[CatalanNumber[Range[30]]]] (* _Harvey P. Dale_, Jan 25 2013 *)

%t CoefficientList[Series[(1 - 2 x - (1 - 4 x)^(1/2))/(2 x (1 - x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 21 2015 *)

%t a[0] := 0; a[n_] := Sum[CatalanNumber[k], {k, 1, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 14 2017 *)

%o (PARI) Vec((1-2*x-(1-4*x)^(1/2))/(2*x*(1-x))) \\ _Charles R Greathouse IV_, Feb 11 2011

%o a014138 n = a014138_list !! n

%o a014138_list = scanl1 (+) a000108_list -- _Reinhard Zumkeller_, Mar 01 2013

%o (Python)

%o from __future__ import division

%o A014138_list, b, s = [0], 1, 0

%o for n in range(1,10**2):

%o s += b

%o A014138_list.append(s)

%o b = b*(4*n+2)//(n+2) # _Chai Wah Wu_, Jan 28 2016

%Y Cf. A000108, A002476, A005554, A068231, A095930, A137614, A155587.

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_

%E Edited by _Max Alekseyev_, Sep 13 2009 (including adding an initial 0)

%E Definition edited by _N. J. A. Sloane_, Oct 03 2009

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Last modified January 16 15:31 EST 2019. Contains 319195 sequences. (Running on oeis4.)