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A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).

%I A128722 #10 Jul 22 2017 08:38:43
%S A128722 1,0,1,2,0,1,6,3,0,1,22,9,4,0,1,84,35,12,5,0,1,334,138,49,15,6,0,1,
%T A128722 1368,563,198,64,18,7,0,1,5734,2352,825,264,80,21,8,0,1,24480,10015,
%U A128722 3504,1121,336,97,24,9,0,1,106086,43308,15123,4833,1452,414,115,27,10,0,1
%N A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).
%C A128722 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 its steps.
%C A128722 T(n,0) = A128723(n).
%C A128722 Row sums yield A002212.
%C A128722 Sum_{k=0..n} k*T(n,k) = A033321(n).
%H A128722 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 A128722 G.f.: (1-z+zg)/(1+z-zg-tz), where g = 1+zg^2+z(g-1) = (1-z-sqrt(1-6z+5z^2))/(2z).
%e A128722 T(3,1)=3 because we have (UD)UUDD, (UD)UUDL and UUDD(UD) (the hills are shown between parentheses).
%e A128722 Triangle starts:
%e A128722    1;
%e A128722    0,  1;
%e A128722    2,  0,  1;
%e A128722    6,  3,  0,  1;
%e A128722   22,  9,  4,  0,  1;
%e A128722   84, 35, 12,  5,  0,  1;
%p A128722 g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z+z*g)/(1+z-z*g-t*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A128722 Cf. A002212, A033321, A128723.
%K A128722 nonn,tabl
%O A128722 0,4
%A A128722 _Emeric Deutsch_, Mar 30 2007