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A128741
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the x-axis (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, 2, 1, 6, 3, 1, 20, 11, 4, 1, 72, 42, 17, 5, 1, 274, 166, 72, 24, 6, 1, 1086, 675, 307, 111, 32, 7, 1, 4438, 2809, 1322, 506, 160, 41, 8, 1, 18570, 11913, 5752, 2296, 775, 220, 51, 9, 1, 79174, 51319, 25274, 10418, 3692, 1127, 292, 62, 10, 1, 342738, 223977, 112054
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row sums yield A002212. T(n,1)=2*A002212(n-1) for n>=2 (obvious: the path of semilength n with exactly one return are of the form UPD and UPL, where P is a path of semilength n-1). Sum(k*T(n,k),k=1..n)=A128742(n).
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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.=(1-tz+tzg)/(1-tzg)-1, where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z). Column k has g.f.=z^k*g^(k-1)*(2g-1).
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EXAMPLE
| T(4,3)=4 because we have U(D)U(D)UUD(D), U(D)U(D)UUD(L), U(D)UUD(D)U(D) and UUD(D)U(D)U(D) (the return steps to the x-axis are shown between parentheses).
Triangle starts:
1;
2,1;
6,3,1;
20,11,4,1;
72,42,17,5,1;
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MAPLE
| g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-t*z+t*z*g)/(1-t*z*g)-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n), n=1..11) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od;
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CROSSREFS
| Cf. A002212, A128742.
Sequence in context: A180281 A187888 A092392 * A175757 A060539 A163269
Adjacent sequences: A128738 A128739 A128740 * A128742 A128743 A128744
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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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