A128731 - OEIS

%I #12 Dec 19 2015 18:02:22

%S 1,1,2,1,5,5,14,21,1,42,84,11,132,330,80,1,429,1287,484,19,1430,5005,

%T 2639,210,1,4862,19448,13468,1780,29,16796,75582,65688,12852,450,1,

%U 58786,293930,310080,83334,5065,41,208012,1144066,1428306,500346,46640

%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DL's (n>=0; 0<=k<=floor(n/2)).

%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 the path is defined to be the number of its steps.

%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.: G=G(t,z) satisfies z^2*G^3-z(2-z)G^2+(1-tz^2)G-1+z=0.

%F Row n has 1+floor(n/2) terms.

%F Row sums yield the sequence A002212.

%F T(n,0) = A000108 (the Catalan numbers).

%F T(n,1) = binomial(2n-1,n-2) = A002054(n-1).

%F Sum_{k=0..floor(n/2)} k*T(n,k) = A128732(n).

%e T(3,1)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the other 5 paths of semilength 3 are Dyck paths which, obviously, contain no DL's).

%e Triangle starts:

%e     1;

%e     1;

%e     2,   1;

%e     5,   5;

%e    14,  21,  1;

%e    42,  84, 11;

%e   132, 330, 80, 1;

%p eq:=z^2*G^3-z*(2-z)*G^2+(1-t*z^2)*G-1+z=0:

%p G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,17)):

%p for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n)) od:

%p for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od;

%p # yields sequence in triangular form

%Y Cf. A000108, A002054, A002212, A128732.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Mar 31 2007