0,3

Also, with offset 1, triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and ending at the point (2k,0) (1<=k<=n). 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. For example, T(3,2)=4 because we have UDUUDL, UUUDLD, UUDUDL, and UUUDDL.

Sum of row n = A002212(n+1). T(n,n)=Catalan(n+1) (A000108).

Sum(k*T(n,k),k=1..n)=A026388(n). Mirror image of A126181.

E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths (in preparation).

With offset 1, T(n,k)=c(k)*binom(n-1,k-1), where c(j)=binom(2j,j)/(j+1) is a Catalan number (A000108). G.f.=G-1, where G=G(t,z) satisfies G=1+tzG^2+z(G-1)

Triangle begins:

1;

1,2;

1,4,5;

1,6,15,14;

1,8,30,56,42;

T:=(n, k)->binomial(2*k+2, k+1)*binomial(n, k)/(k+2): for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Cf. A002212, A000108, A026388, A126181.

Sequence in context: A050166 A124959 A081281 this_sequence A121289 A134248 A080935

Adjacent sequences: A108195 A108196 A108197 this_sequence A108199 A108200 A108201

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2005, Mar 30 2007

Edited by njas at the suggestion of Andrew Pewe, Jun 16 2007