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Search: id:A128749
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| A128749 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps. |
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+0
2
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| 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 14, 12, 9, 0, 1, 44, 53, 25, 14, 0, 1, 150, 196, 132, 44, 20, 0, 1, 520, 777, 555, 269, 70, 27, 0, 1, 1850, 3064, 2486, 1260, 485, 104, 35, 0, 1, 6696, 12233, 10902, 6264, 2496, 804, 147, 44, 0, 1, 24602, 49096, 47955, 30108, 13600
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums yield A002212. T(n,0)=A128750(n). Sum(k*T(n,k),k=0..n)=A085362(n-1).
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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+tz^2-z^2)G+1-z=0.
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EXAMPLE
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T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD, and UUD(U)DL (the ascents of length 1 are shown between parentheses).
Triangle starts:
1;
0,1;
2,0,1;
4,5,0,1;
14,12,9,0,1;
44,53,25,14,0,1;
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MAPLE
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eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A128750, A085362.
Sequence in context: A098689 A139435 A077909 this_sequence A106579 A016584 A112899
Adjacent sequences: A128746 A128747 A128748 this_sequence A128750 A128751 A128752
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KEYWORD
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tabl,nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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