A128722 - OEIS

%I #10 Jul 22 2017 08:38:43

%S 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 1368,563,198,64,18,7,0,1,5734,2352,825,264,80,21,8,0,1,24480,10015,

%U 3504,1121,336,97,24,9,0,1,106086,43308,15123,4833,1452,414,115,27,10,0,1

%N 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 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 T(n,0) = A128723(n).

%C Row sums yield A002212.

%C Sum_{k=0..n} k*T(n,k) = A033321(n).

%H 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 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 T(3,1)=3 because we have (UD)UUDD, (UD)UUDL and UUDD(UD) (the hills are shown between parentheses).

%e Triangle starts:

%e    1;

%e    0,  1;

%e    2,  0,  1;

%e    6,  3,  0,  1;

%e   22,  9,  4,  0,  1;

%e   84, 35, 12,  5,  0,  1;

%p 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 Cf. A002212, A033321, A128723.

%K nonn,tabl

%O 0,4

%A _Emeric Deutsch_, Mar 30 2007