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

0,3

Comments

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

Links

Table of n, a(n) for n=0..59.

Formula

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

Example

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.

Sequence in context: A092107 A114489 A101974 * A132893 A273896 A344363

Adjacent sequences: A097604 A097605 A097606 * A097608 A097609 A097610

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 30 2004

Status

approved