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A128731
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DL's (n>=0; 0<=k<=floor(n/2)). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
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1
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1, 1, 2, 1, 5, 5, 14, 21, 1, 42, 84, 11, 132, 330, 80, 1, 429, 1287, 484, 19, 1430, 5005, 2639, 210, 1, 4862, 19448, 13468, 1780, 29, 16796, 75582, 65688, 12852, 450, 1, 58786, 293930, 310080, 83334, 5065, 41, 208012, 1144066, 1428306, 500346, 46640
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n has 1+floor(n/2) terms. Row sums yield the sequence A002212. T(n,0)=A000108 (the Catalan numbers). T(n,1)=binom(2n-1,n-2)=A002054(n-1). Sum(k*T(n,k), k=0..floor(n/2))=A128732.
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| G.f.=G=G(t,z) satisfies z^2*G^3-z(2-z)G^2+(1-tz^2)G-1+z=0.
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EXAMPLE
| T(3,1)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the other 5 paths of semilength 3 are Dyck paths which, obviously, contain no DL's).
Triangle starts:
1;
1;
2,1;
5,5;
14,21,1;
42,84,11;
132,330,80,1;
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MAPLE
| eq:=z^2*G^3-z*(2-z)*G^2+(1-t*z^2)*G-1+z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A002212, A000108, A002054, A128732.
Sequence in context: A096976 A052547 A119245 * A129157 A086905 A167638
Adjacent sequences: A128728 A128729 A128730 * A128732 A128733 A128734
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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