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A097607
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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).
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1
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums are the Catalan numbers (A000108) Column 0 is A014137 (partial sums of Catalan numbers). Column 1 is A001453 (Catalan numbers -1).
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FORMULA
| G.f.=(1-z+zC-tzC)/[(1-z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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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.
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CROSSREFS
| Cf. A000108, A014137, A001453.
Sequence in context: A092107 A114489 A101974 * A132893 A163240 A091958
Adjacent sequences: A097604 A097605 A097606 * A097608 A097609 A097610
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 30 2004
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