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A128728 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDL's (n >= 0; 0 <= k <= floor((n+1)/2)).

%I A128728 #19 Jul 06 2023 01:56:53
%S A128728 1,1,2,1,6,4,20,16,71,64,2,262,261,20,994,1084,141,3852,4572,854,7,
%T A128728 15183,19520,4772,112,60686,84139,25416,1128,245412,365404,131270,
%U A128728 9120,30,1002344,1596420,664004,64790,660,4129012,7008544,3309336,422928
%N A128728 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDL's (n >= 0; 0 <= k <= floor((n+1)/2)).
%C A128728 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of steps in it.
%C A128728 Row n has 1 + floor((n+1)/3) terms.
%C A128728 Row sums yield A002212.
%C A128728 T(n,0) = A128729(n).
%C A128728 Sum_{k>=0} k*T(n,k) = A128730(n).
%C A128728 Apparently, T(3k-1,k) = binomial(3k-1,k)/(3k-1) = A006013(k-1).
%H A128728 E. Deutsch, E. Munarini, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%H A128728 Helmut Prodinger, <a href="https://arxiv.org/abs/2203.10516">Skew Dyck paths without up-down-left</a>, arXiv:2203.10516 [math.CO], 2022.
%H A128728 Yuxuan Zhang and Yan Zhuang, <a href="https://arxiv.org/abs/2306.15778">A subfamily of skew Dyck paths related to k-ary trees</a>, arXiv:2306.15778 [math.CO], 2023.
%F A128728 G.f.: G = G(t,z) satisfies z^2*G^3 - z(2-z)G^2 + (1-z^2)G - 1 + z + z^2 - tz^2 = 0.
%e A128728 T(3,1)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL.
%e A128728 Triangle starts:
%e A128728     1;
%e A128728     1;
%e A128728     2,   1;
%e A128728     6,   4;
%e A128728    20,  16;
%e A128728    71,  64,   2;
%e A128728   262, 261,  20;
%p A128728 eq:=z^2*G^3-z*(2-z)*G^2+(1-z^2)*G-1+z+z^2-t*z^2=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((n+1)/3)) od; # yields sequence in triangular form
%Y A128728 Cf. A002212, A006013, A128729, A128730.
%K A128728 nonn,tabf
%O A128728 0,3
%A A128728 _Emeric Deutsch_, Mar 31 2007