A143949 Triangle read by rows: T(n,k) is the number of n-Dyck paths containing k odd-length descents to ground level (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 5, 0, 1, 10, 17, 7, 7, 0, 1, 32, 46, 34, 10, 9, 0, 1, 100, 155, 94, 55, 13, 11, 0, 1, 329, 502, 335, 154, 80, 16, 13, 0, 1, 1101, 1701, 1110, 580, 226, 109, 19, 15, 0, 1, 3761, 5820, 3865, 1960, 898, 310, 142, 22, 17, 0, 1, 13035, 20251

Offset: 0

Author

Emeric Deutsch, Oct 05 2008

Keywords

• nonn
• tabl

Comments

Row sums are the Catalan numbers (A000108).

T(0,n)=A033297(n).

Sum(k*T(n,k),k=0..n)=A000957(n+2) (the Fine numbers).

The case of even-length descents to ground level is considered in A111301.

Examples

			T(4,2) = 5 because we have U(D)U(D)UUDD, U(D)UUDDU(D), U(D)UUU(DDD), UUDDU(D)U(D) and UUU(DDD)U(D) (the odd-length descents to ground level are shown between parentheses).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
4,4,5,0,1;
10,17,7,7,0,1;
		

Crossrefs

Cf. A000108, A033297, A000957, A111301.

Programs

Maple

C:=((1-sqrt(1-4*z))*1/2)/z: G:=1/(1-z*(t+z*C)/(1-z^2*C^2)): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

Formula

G.f.: G(s,z) = 1/[1-z(t+zC)/(1-z^2*C^2)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.

The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.