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A129159 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having abscissa of the first return to the x-axis equal to 2k (1 <= k <= n).

%I A129159 #11 Jul 23 2017 12:18:02
%S A129159 1,2,1,4,4,2,11,9,11,5,37,21,31,34,14,138,59,76,116,112,42,544,198,
%T A129159 198,315,448,384,132,2220,743,599,825,1358,1758,1353,429,9286,2964,
%U A129159 2091,2345,3724,5922,6963,4862,1430,39588,12251,8026,7604,10388,17304,25872
%N A129159 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having abscissa of the first return to the x-axis equal to 2k (1 <= k <= n).
%C A129159 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 the path is defined to be the number of its steps.
%C A129159 Row sums yield A002212.
%C A129159 T(n,1) = 1 + A002212(n-1) (indeed, the path U^nDL^(n-1) and the paths UDP, where P is a skew Dyck path of semilength n-1).
%C A129159 T(n,n) = binomial(2n-2,n-1)/n = A000108(n-1) (the Catalan numbers).
%H A129159 E. Deutsch, E. Munarini, 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.
%F A129159 Sum_{k=1..n} k*T(n,k) = A129160(n).
%F A129159 G.f.: tzhg + z(h-1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2)) and h = 1 + tzh^2 + z(h-1) (h = h(t,z) is the g.f. for skew Dyck paths according to the semi-abscissa of the last point on the x-axis and semilength; see A108198).
%e A129159 T(3,2)=4 because we have UUDDUD, UUUDLD, UUDUDL and UUUDDL.
%e A129159 Triangle starts:
%e A129159    1;
%e A129159    2,  1;
%e A129159    4,  4,  2;
%e A129159   11,  9, 11,  5;
%e A129159   37, 21, 31, 34, 14;
%p A129159 g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: h:=(1-z-sqrt(z^2-2*z+1+4*t*z^2-4*t*z))/2/t/z: G:=t*z*h*g+z*(h-1): Gser:=simplify(series(G,z=0,14)): for n from 1 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y A129159 Cf. A000108, A002212, A108198, A129160.
%K A129159 nonn,tabl
%O A129159 1,2
%A A129159 _Emeric Deutsch_, Apr 03 2007