A097607 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having leftmost valley at altitude k (if path has no valleys, then this altitude is considered to be 0).

1, 1, 2, 4, 1, 9, 4, 1, 23, 13, 5, 1, 65, 41, 19, 6, 1, 197, 131, 67, 26, 7, 1, 626, 428, 232, 101, 34, 8, 1, 2056, 1429, 804, 376, 144, 43, 9, 1, 6918, 4861, 2806, 1377, 573, 197, 53, 10, 1, 23714, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1, 82500, 58785, 35072

Offset: 0

Author

Emeric Deutsch, Aug 30 2004

Keywords

• nonn
• tabf

Comments

Row sums are the Catalan numbers (A000108) Column 0 is A014137 (partial sums of Catalan numbers). Column 1 is A001453 (Catalan numbers -1).

Examples

			Triangle starts:
1;
1;
2;
4,1;
9,4,1;
23,13,5,1;
65,41,19,6,1;
T(4,1)=4 because we have UU(DU)DDUD, UU(DU)DUDD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the first valleys, all at altitude 1, are shown between parentheses.
		

Crossrefs

Cf. A000108, A014137, A001453.

Formula

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