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Search: id:A128753
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| A128753 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n>=0; 0<=k<=n-2 for 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 a path is defined to be the number of its steps. |
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+0
1
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| 1, 1, 3, 9, 1, 31, 4, 1, 113, 19, 4, 1, 431, 86, 21, 4, 1, 1697, 393, 101, 23, 4, 1, 6847, 1800, 492, 116, 25, 4, 1, 28161, 8279, 2388, 596, 131, 27, 4, 1, 117631, 38218, 11603, 3032, 705, 146, 29, 4, 1, 497665, 177013, 56407, 15403, 3732, 819, 161, 31, 4, 1
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Rows 0 and 1 have one term each; row n has n-1 terms (n>=2). Row sums yield A002212. T(n,0)=A052709(n+1). Sum(k*T(n,k),k=0..n-2)=A026376(n-2).
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REFERENCES
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E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths (in preparation).
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FORMULA
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G.f.=G=G(t,z) satisfies z(1+z-tz)G^2-(1-tz)G+1-tz=0. G=C((1+z-tz)/(1-tz)), where C(z)=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(4,1)=4 because we have (UDUDU)UDD, (UDUDU)UDL, U(UDUDU)DD, and U(UDUDU)DL (the subwords UDUDU are shown between parentheses).
Triangle starts
1;
1;
3;
9,1;
31,4,1;
113,19,4,1;
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MAPLE
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C:=z->(1-sqrt(1-4*z))/2/z: G:=C(z*(1+z-t*z)/(1-t*z)): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A052709, A026376.
Sequence in context: A126177 A128733 A128724 this_sequence A016048 A021259 A114875
Adjacent sequences: A128750 A128751 A128752 this_sequence A128754 A128755 A128756
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2007
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