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A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

%I A025566 #47 Jun 25 2023 13:22:35
%S A025566 1,1,1,3,8,22,61,171,483,1373,3923,11257,32418,93644,271219,787333,
%T A025566 2290200,6673662,19478091,56930961,166613280,488176938,1431878079,
%U A025566 4203938697,12353600427,36331804089,106932444885,314946659951,928213563878
%N A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.
%C A025566 a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - _David Callan_, Jun 07 2006
%C A025566 Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - _Gary W. Adamson_, Jan 04 2008
%H A025566 Michael De Vlieger, <a href="/A025566/b025566.txt">Table of n, a(n) for n = 0..2100</a>
%H A025566 Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, <a href="https://arxiv.org/abs/1911.03119">Pattern distributions in Dyck paths with a first return decomposition constrained by height</a>, arXiv:1911.03119 [math.CO], 2019.
%H A025566 C. Dalfó, M. A. Fiol, and N. López, <a href="https://arxiv.org/abs/2007.09639">New results for the Mondrian art problem</a>, arXiv:2007.09639 [math.CO], 2020.
%H A025566 D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i2p33">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
%H A025566 Christian Krattenthaler, Daniel Yaqubi, <a href="https://arxiv.org/abs/1802.05990">Some determinants of path generating functions, II</a>, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
%H A025566 Donatella Merlini, Massimo Nocentini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Merlini/merlini5.html">Algebraic Generating Functions for Languages Avoiding Riordan Patterns</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
%F A025566 G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - _Emeric Deutsch_, Aug 16 2002
%F A025566 Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - _Gary W. Adamson_, Sep 01 2007
%F A025566 a(n) is the sum of the (n-2)-th row of triangle A131816. - _Gary W. Adamson_, Sep 01 2007
%F A025566 D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - _R. J. Mathar_, Sep 15 2020
%p A025566 seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%t A025566 CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* _Harvey P. Dale_, Jun 12 2016 *)
%o A025566 (GAP) List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # _Muniru A Asiru_, Mar 09 2019
%Y A025566 First differences of A026135. Row sums of triangle A026105.
%Y A025566 Pairwise sums of A005727. Column k=2 in A115990.
%Y A025566 Cf. A001791, A132816.
%Y A025566 Cf. A136537.
%K A025566 nonn
%O A025566 0,4
%A A025566 _Clark Kimberling_