A128741 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the x-axis (1 <= k <= n).

1, 2, 1, 6, 3, 1, 20, 11, 4, 1, 72, 42, 17, 5, 1, 274, 166, 72, 24, 6, 1, 1086, 675, 307, 111, 32, 7, 1, 4438, 2809, 1322, 506, 160, 41, 8, 1, 18570, 11913, 5752, 2296, 775, 220, 51, 9, 1, 79174, 51319, 25274, 10418, 3692, 1127, 292, 62, 10, 1, 342738, 223977, 112054

Offset: 1

Author

Emeric Deutsch, Mar 31 2007

Keywords

• tabl
• nonn

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.

Row sums yield A002212.

T(n,1) = 2*A002212(n-1) for n >= 2 (obvious: the path of semilength n with exactly one return are of the form UPD and UPL, where P is a path of semilength n-1).

Sum_{k=1..n} k*T(n,k) = A128742(n).

Examples

			T(4,3)=4 because we have U(D)U(D)UUD(D), U(D)U(D)UUD(L), U(D)UUD(D)U(D) and UUD(D)U(D)U(D) (the return steps to the x-axis are shown between parentheses).
Triangle starts:
   1;
   2,  1;
   6,  3,  1;
  20, 11,  4,  1;
  72, 42, 17,  5,  1;
		

Links

• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Crossrefs

Cf. A002212, A128742.

Programs

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-t*z+t*z*g)/(1-t*z*g)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n),n=1..11) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od;

Formula

G.f.: (1 - tz + tzg)/(1 - tzg) - 1, where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).

Column k has g.f. z^k*g^(k-1)*(2g-1).