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Search: id:A128738
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| A128738 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k DD's (0<=k<=n-1 for n>=1). 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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+0
3
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| 1, 2, 1, 5, 4, 1, 14, 14, 7, 1, 41, 51, 33, 11, 1, 124, 188, 145, 69, 16, 1, 386, 690, 627, 362, 131, 22, 1, 1230, 2529, 2655, 1790, 821, 230, 29, 1, 3992, 9283, 11033, 8533, 4610, 1719, 379, 37, 1, 13150, 34135, 45257, 39435, 24434, 10957, 3361, 593, 46, 1
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n has n terms (n>=1). Row sums yield the sequence A002212. T(n,0)=A128739(n). Sum(k*T(n,k), k=0..n-1)=A128740(n).
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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^2*G^3-z(1-t)(1-z)G^2-(1-z)(1-3z+tz)G+(1-z)^2=0.
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EXAMPLE
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T(3,1)=4 because we have UDUUDD, UUDDUD, UUDUDD, and UUUDDL.
Triangle starts:
1;
1;
2,1;
5,4,1;
14,14,7,1;
41,51,33,11,1;
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MAPLE
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eq:=z^2*G^3-z*(1-z)*(1-t)*G^2-(1-z)*(1-3*z+z*t)*G+(1-z)^2=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A128739, A128740.
Sequence in context: A096164 A104710 A039598 this_sequence A126181 A104259 A137650
Adjacent sequences: A128735 A128736 A128737 this_sequence A128739 A128740 A128741
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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), Mar 31 2007
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