The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A086615 Antidiagonal sums of triangle A086614. 9
 1, 2, 4, 8, 17, 38, 89, 216, 539, 1374, 3562, 9360, 24871, 66706, 180340, 490912, 1344379, 3701158, 10237540, 28436824, 79288843, 221836402, 622599625, 1752360040, 4945087837, 13988490338, 39658308814, 112666081616 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Partial sums of the Motzkin sequence (A001006). - Emeric Deutsch, Jul 12 2004 a(n) = number of distinct ordered trees obtained by branch-reducing the ordered trees on n+1 edges. - David Callan, Oct 24 2004 a(n)= the number of consecutive horizontal steps at height 0 of all Motzkin paths from (0,0) to (n,0) starting with a horizontal step. - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007 Equals row sums of triangle A136788 - Gary W. Adamson, Jan 21 2008 The subsequence of prime partial sums of the Motzkin sequence begins: 2, 17, 89, no more through a(27). [From Jonathan Vos Post, Feb 11 2010] This sequence (with offset 1 instead of 0) occurs in Section 7 of K. Grygiel, P. Lescanne (2015), see g.f. N. - N. J. A. Sloane, Nov 09 2015 Also number of plain (untyped) normal forms of lambda-terms (terms that cannot be further beta-reduced.) [Bendkowski et al., 2016]. - N. J. A. Sloane, Nov 22 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 P. Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6 Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682, 2016 K. Grygiel, P. Lescanne, A natural counting of lambda terms, SOFSEM 2016. Preprint 2015 FORMULA G.f.: A(x) = 1/(1-x)^2 + x^2*A(x)^2. a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, 2k+1)binomial(2k, k)/(k+1)} - Paul Barry, Nov 29 2004 a(n) = n + 1 + sum_k a(k-1)a(n-k-1), starting from a(n)=0 for n negative. - Henry Bottomley, Feb 22 2005 a(n)=sum{k=0..n, sum{j=0..n-k, C(j)C(n-k, 2j)}}; - Paul Barry, Aug 19 2005 G.f.: c(x^2/(1-x)^2)/(1-x)^2, c(x) the g.f. of A000108; a(n)=sum{k=0..floor(n/2), C(n+1,n-2k)C(k)}; - Paul Barry, May 31 2006 Binomial transform of doubled Catalan sequence 1,1,1,1,2,2,5,5,14,14,... - Paul Barry, Nov 17 2005 Row sums of Pascal-Catalan triangle A086617. - Paul Barry, Nov 17 2005 g(z)=(1-z-sqrt(1-2z-3z^2))/(2z-2z^2)/z - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007, corrected by Vaclav Kotesovec, Feb 13 2014 Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n+4)*a(n-2) +3*(n-1)*a(n-3)=0. - R. J. Mathar, Nov 30 2012 a(n) ~ 3^(n+5/2) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014 EXAMPLE a(0)=1, a(1)=2, a(2)=3+1=4, a(3)=4+4=8, a(4)=5+10+2=17, a(5)=6+20+12=38, are upward antidiagonal sums of triangle A086614: {1}, {2,1}, {3,4,2}, {4,10,12,5}, {5,20,42,40,14}, {6,35,112,180,140,42}, ... For example with n=2, the 5 ordered trees (A000108) on 3 edges are |...|..../\.../\.../|\.. |../.\..|......|........ |....................... Suppressing nonroot vertices of outdegree 1 (branch-reducing) yields |...|..../\.../\../|\.. .../.\................. of which 4 are distinct. So a(2)=4. a(4)=8 because we have HHHH, HHUD, HUDH, HUHD MATHEMATICA CoefficientList[Series[(1-x-Sqrt[1-2*x-3*x^2])/(2*x-2*x^2)/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *) CROSSREFS Cf. A086614 (triangle), A086616 (row sums). Cf. A001006. Cf. A136788. Sequence in context: A257300 A229202 A003007 * A081124 A090901 A101516 Adjacent sequences:  A086612 A086613 A086614 * A086616 A086617 A086618 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 24 2003 EXTENSIONS Edited by N. J. A. Sloane, Oct 16 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 02:18 EST 2020. Contains 338921 sequences. (Running on oeis4.)