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A128744
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1<=k<=n). 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
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1, 1, 2, 3, 3, 4, 10, 10, 8, 8, 36, 36, 29, 20, 16, 137, 137, 111, 78, 48, 32, 543, 543, 442, 315, 200, 112, 64, 2219, 2219, 1813, 1306, 848, 496, 256, 128, 9285, 9285, 7609, 5527, 3649, 2200, 1200, 576, 256, 39587, 39587, 32521, 23779, 15901, 9802, 5552, 2848
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums yield A002212. T(n,1)=A002212(n-1). T(n,2)=A002212(n-1) for n>=3. Sum(k*T(n,k),k=1..n)=A039919(n+1).
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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.= tzg/(1-tz-tzg), where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z).
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EXAMPLE
| T(3,3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
1;
1,2;
3,3,4;
10,10,8,8;
36,36,29,20,16;
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MAPLE
| g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z*g/(1-t*z-t*z*g): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A002212, A039919.
Sequence in context: A206455 A111574 A173590 * A118963 A127641 A106821
Adjacent sequences: A128741 A128742 A128743 * A128745 A128746 A128747
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KEYWORD
| tabl,nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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