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A128744 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1 <= k <= n).

%I A128744 #10 Jul 23 2017 12:17:20
%S A128744 1,1,2,3,3,4,10,10,8,8,36,36,29,20,16,137,137,111,78,48,32,543,543,
%T A128744 442,315,200,112,64,2219,2219,1813,1306,848,496,256,128,9285,9285,
%U A128744 7609,5527,3649,2200,1200,576,256,39587,39587,32521,23779,15901,9802,5552,2848
%N A128744 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1 <= k <= n).
%C A128744 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 A128744 Row sums yield A002212.
%H A128744 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 A128744 T(n,1) = A002212(n-1).
%F A128744 T(n,2) = A002212(n-1) for n >= 3.
%F A128744 Sum_{k=1..n} k*T(n,k) = A039919(n+1).
%F A128744 G.f.: t*z*g/(1 - t*z - t*z*g), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
%e A128744 T(3,3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
%e A128744 Triangle starts:
%e A128744    1;
%e A128744    1,  2;
%e A128744    3,  3,  4;
%e A128744   10, 10,  8,  8;
%e A128744   36, 36, 29, 20, 16;
%p A128744 g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z*g/(1-t*z-t*z*g): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(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 A128744 Cf. A002212, A039919.
%K A128744 tabl,nonn
%O A128744 1,3
%A A128744 _Emeric Deutsch_, Mar 31 2007