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%I M0801 #54 Feb 05 2022 14:33:31
%S 1,2,3,6,13,30,72,178,450,1158,3023,7986,21309,57346,155469,424206,
%T 1164039,3210246,8893161,24735666,69051303,193399578,543310782,
%U 1530523638,4322488212,12236130298,34713220977,98677591278
%N Sums of successive Motzkin numbers.
%C The Donaghey reference shows that a(n) is the number of n-vertex binary trees such that for each non-root vertex that is incident to exactly two edges, these two edges have opposite slope. It also notes that these trees correspond to Dyck n-paths (A000108) containing no DUDUs and no subpaths of the form UUPDD with P a nonempty Dyck path. For example, a(3)=3 counts UUDUDD, UDUUDD, UUDDUD. - _David Callan_, Sep 25 2006
%C Hankel transform of the sequence starting with 2 appears to be 3, 4, 5, 6, 7, ... _Gary W. Adamson_, May 27 2011
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005554/b005554.txt">Table of n, a(n) for n = 1..1000</a>
%H R. Donaghey, <a href="http://dx.doi.org/10.1016/0095-8956(80)90045-3">Automorphisms on Catalan trees and bracketing</a>, J. Combin. Theory, Series B, 29 (1980), 75-90.
%H Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%F Inverse binomial transform of A014138: (1, 3, 8, 22, 64, 196, ...). - _Gary W. Adamson_, Nov 23 2007
%F D-finite with recurrence (n + 1)*a(n) = 2*n*a(n - 1) + (3*n - 9)*a(n - 2).
%F G.f.: (x+x^2)*M(x) where M(x)=(1 - x - (1 - 2*x - 3*x^2)^(1/2))/(2*x^2) is the g.f. for the Motzkin numbers A001006. - _David Callan_, Sep 25 2006
%F a(n) = (-1)^n*2*hypergeometric([2-n,5/2],[4],4), for n>1. - _Peter Luschny_, Aug 15 2012
%F a(n) ~ 2*3^(n-1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F a(n) = (2*Sum_{j=0..(n+2)/2} (binomial(n,j)*binomial(n-j+1,n-2*j+2)))/n. - _Vladimir Kruchinin_, Oct 04 2015
%t Rest[CoefficientList[Series[(x+x^2)*(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o (Maxima)
%o a(n):=(2*sum(binomial(n,j)*binomial(n-j+1,n-2*j+2),j,0,(n+2)/2))/n; /* _Vladimir Kruchinin_, Oct 04 2015 */
%o (PARI) a(n) = sum(k=0, (n+2)/2, 2*(binomial(n,k)*binomial(n-k+1,n-2*k+2)/n));
%o vector(40, n, if(n==1, 1, a(n-1))) \\ _Altug Alkan_, Oct 04 2015
%Y Enumerates the branch-reduced trees encoded by A080981. Cf. A001006.
%Y First differences are in A102071.
%Y Cf. A014138.
%Y A diagonal of A059346.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Jul 10 2000
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