A129157 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive non-Dyck factors (n>=0; 0<=k<=floor((n+1)/3)).

1, 1, 2, 1, 5, 5, 14, 22, 42, 94, 1, 132, 400, 11, 429, 1709, 81, 1430, 7351, 503, 1, 4862, 31857, 2851, 17, 16796, 139100, 15297, 176, 58786, 611781, 79228, 1440, 1, 208012, 2709230, 400694, 10259, 23, 742900, 12075248, 1993226, 66774, 307, 2674440

Offset: 0

Author

Emeric Deutsch, Apr 02 2007

Keywords

• nonn
• tabf

Comments

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. A primitive non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.

Examples

			T(3,1) = 5 because we have UD(UUDL), (UUUDLD), (UUDUDL), (UUUDDL) and (UUUDLL);
T(5,2) = 1 because we have (UUUDLD)(UUDL) (the primitive non-Dyck factors are shown between parentheses).
Triangle starts:
    1;
    1;
    2,   1;
    5,   5;
   14,  22;
   42,  94,  1;
  132, 400, 11;
		

Links

• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2192-2203

Crossrefs

Cf. A000108, A002212, A129154, A129158.

Programs

Maple

G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(1+t*z+(1-t)*sqrt(1-4*z)+t*sqrt(1-6*z+5*z^2)):
Gser:=simplify(series(G,z=0,18)):
for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) od:
for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..floor((n+1)/3)) od;
# yields sequence in triangular form

Formula

G.f.: G(t,z) = [1+tz(g-1)]/[1-tz(g-C)-zC], where g=1+zg^2+z(g-1) = [1-z-sqrt(1- 6z+5z^2)]/(2z) and C=1+zC^2=[1-sqrt(1-4z)]/(2z) is the Catalan function.

Row n has 1+floor((n+1)/3) terms (n>=1).

Row sums yield A002212.

T(n,0) = binomial(2*n,n)/(n+1) = A000108(n) (the Catalan numbers).

Sum_{k>=0} k*T(n,k) = A129158(n).