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A128733
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k LD's (n>=0; 0<=k<=floor((n-1)/2)).
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1
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1, 3, 9, 1, 28, 8, 90, 46, 1, 297, 231, 15, 1001, 1079, 138, 1, 3432, 4823, 1006, 24, 11934, 20944, 6388, 320, 1, 41990, 89148, 37026, 3170, 35, 149226, 374034, 201210, 26130, 635, 1, 534888, 1552661, 1042492, 189959, 8170, 48, 1931540, 6393310
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OFFSET
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0,2
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COMMENTS
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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.
Row n has ceiling(n/2) terms (n >= 1).
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LINKS
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E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
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T(n,0) = 3(2n)!/((n+2)!(n-1)!) = A000245(n) (n >= 1).
Sum_{k=0..floor((n-1)/2)} k*T(n,k) = A128734(n).
G.f.: G = G(t,z) satisfies tz^2*G^3 - (t-1)z^2*G^2 - (1 - 3z + 2z^2)G + (1-z)^2 = 0.
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EXAMPLE
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T(3,1)=1 because we have UUUDLD.
Triangle starts:
1;
1;
3;
9, 1;
28, 8;
90, 46, 1;
297, 231, 15;
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MAPLE
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eq:=t*z^2*G^3-(t-1)*z^2*G^2-(1-3*z+2*z^2)*G+(1-z)^2=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..floor((n-1)/2)) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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