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%I #54 Mar 07 2023 11:06:21
%S 1,2,4,8,17,38,89,216,539,1374,3562,9360,24871,66706,180340,490912,
%T 1344379,3701158,10237540,28436824,79288843,221836402,622599625,
%U 1752360040,4945087837,13988490338,39658308814,112666081616
%N Antidiagonal sums of triangle A086614.
%C Partial sums of the Motzkin sequence (A001006). - _Emeric Deutsch_, Jul 12 2004
%C a(n) is the number of distinct ordered trees obtained by branch-reducing the ordered trees on n+1 edges. - _David Callan_, Oct 24 2004
%C a(n) is 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
%C 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
%C 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
%C If interpreted with offset 2, the INVERT transform is A002026 with offset 1. - _R. J. Mathar_, Nov 02 2021
%H Vincenzo Librandi, <a href="/A086615/b086615.txt">Table of n, a(n) for n = 0..200</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6
%H Maciej Bendkowski, K. Grygiel, and P. Tarau, <a href="http://arxiv.org/abs/1612.07682">Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers</a>, arXiv preprint arXiv:1612.07682, 2016
%H K. Grygiel and P. Lescanne, <a href="http://dx.doi.org/10.1007/978-3-662-49192-8_15">A natural counting of lambda terms</a>, SOFSEM 2016. <a href="http://perso.ens-lyon.fr/pierre.lescanne/PUBLICATIONS/natural_counting.pdf">Preprint 2015</a>
%F G.f.: A(x) = 1/(1-x)^2 + x^2*A(x)^2.
%F a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+1, 2k+1)*binomial(2k, k)/(k+1). - _Paul Barry_, Nov 29 2004
%F 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
%F a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(j)*C(n-k, 2j). - _Paul Barry_, Aug 19 2005
%F From _Paul Barry_, May 31 2006: (Start)
%F G.f.: c(x^2/(1-x)^2)/(1-x)^2, c(x) the g.f. of A000108;
%F a(n) = Sum_{k=0..floor(n/2)} C(n+1,n-2k)*C(k). (End)
%F Binomial transform of doubled Catalan sequence 1,1,1,1,2,2,5,5,14,14,... - _Paul Barry_, Nov 17 2005
%F Row sums of Pascal-Catalan triangle A086617. - _Paul Barry_, Nov 17 2005
%F 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
%F D-finite with recurrence (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
%F a(n) ~ 3^(n+5/2) / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%e 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:
%e {1},
%e {2,1},
%e {3,4,2},
%e {4,10,12,5},
%e {5,20,42,40,14},
%e {6,35,112,180,140,42}, ...
%e For example, with n=2, the 5 ordered trees (A000108) on 3 edges are
%e |...|..../\.../\.../|\..
%e |../.\..|......|........
%e |.......................
%e Suppressing nonroot vertices of outdegree 1 (branch-reducing) yields
%e |...|..../\.../\../|\..
%e .../.\.................
%e of which 4 are distinct. So a(2)=4.
%e a(4)=8 because we have HHHH, HHUD, HUDH, HUHD
%p A086615 := proc(n)
%p option remember;
%p if n <= 3 then
%p 2^n;
%p else
%p 3*(-n-1)*procname(n-1) +(-n+4)*procname(n-2) +3*(n-1)*procname(n-3) ;
%p -%/(n+2) ;
%p end if;
%p end proc:
%p seq(A086615(n),n=0..20) ; # _R. J. Mathar_, Nov 02 2021
%t 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 *)
%Y Cf. A086614 (triangle), A086616 (row sums), A348869 (Seq. Transf.).
%Y Cf. A001006.
%Y Cf. A136788.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 24 2003
%E Edited by _N. J. A. Sloane_, Oct 16 2006
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