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Search: id:A129159
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| A129159 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having abscissa of the first return to the x-axis equal to 2k (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
2
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| 1, 2, 1, 4, 4, 2, 11, 9, 11, 5, 37, 21, 31, 34, 14, 138, 59, 76, 116, 112, 42, 544, 198, 198, 315, 448, 384, 132, 2220, 743, 599, 825, 1358, 1758, 1353, 429, 9286, 2964, 2091, 2345, 3724, 5922, 6963, 4862, 1430, 39588, 12251, 8026, 7604, 10388, 17304, 25872
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums yield A002212. T(n,1)=1+A002212(n-1) (indeed, the path U^nDL^(n-1) and the paths UDP, where P is a skew Dyck path of semilength n-1). T(n,n)=binom(2n-2,n-1)/n = A000108(n-1) (the Catalan numbers). Sum(k*T(n,k),k=1..n)=A129160(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.=tzhg + z(h-1), where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)] and h=1+tzh^2+z(h-1) (h=h(t,z) is the g.f. for skew Dyck paths according to the semi-abscissa of the last point on the x-axis and semilength; see A108198).
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EXAMPLE
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T(3,2)=4 because we have UUDDUD, UUUDLD, UUDUDL and UUUDDL.
Triangle starts:
1;
2,1;
4,4,2;
11,9,11,5;
37,21,31,34,14;
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: h:=(1-z-sqrt(z^2-2*z+1+4*t*z^2-4*t*z))/2/t/z: G:=t*z*h*g+z*(h-1): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 11 do P[n]:=sort(expand(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
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Cf. A002212, A000108, A108198, A129160.
Sequence in context: A135366 A051289 A090802 this_sequence A095830 A101621 A086484
Adjacent sequences: A129156 A129157 A129158 this_sequence A129160 A129161 A129162
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 03 2007
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