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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, 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 (list; graph; refs; listen; history; text; internal format)
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 A163240

Adjacent sequences:  A097604 A097605 A097606 * A097608 A097609 A097610

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 30 2004

STATUS

approved

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Last modified December 13 23:14 EST 2019. Contains 329974 sequences. (Running on oeis4.)