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_Emeric Deutsch (deutsch(AT)duke.poly.edu), _, Aug 30 2004
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.
nonn,tabf,new
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
0,3
G.f.=(1-z+zC-tzC)/[(1-z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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.
nonn,tabf
Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 30 2004
approved