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Search: id:A129157
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| A129157 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive non-Dyck factors (n>=0; 0<=k<=floor((n+1)/3)). 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. A primitive non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path. |
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+0
4
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| 1, 1, 2, 1, 5, 5, 14, 22, 42, 94, 1, 132, 400, 11, 429, 1709, 81, 1430, 7351, 503, 1, 4862, 31857, 2851, 17, 16796, 139100, 15297, 176, 58786, 611781, 79228, 1440, 1, 208012, 2709230, 400694, 10259, 23, 742900, 12075248, 1993226, 66774, 307, 2674440
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row n has 1+floor((n+1)/3) terms (n>=1). Row sums yield A002212. T(n,0)=binom(2n,n)/(n+1)=A000108(n) (the Catalan numbers). Sum(k*T(n,k),k>=0)=A129158(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.=G(t,z)=[1+tz(g-1)]/[1-tz(g-C)-zC], where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z) and C=1+zC^2=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(3,1)=5 because we have UD(UUDL), (UUUDLD), (UUDUDL), (UUUDDL) and (UUUDLL); T(5,2)=1 because we have (UUUDLD)(UUDL) (the primitive non-Dyck factors are shown between parentheses).
Triangle starts:
1;
1;
2,1;
5,5;
14,22;
42,94,1;
132,400,11;
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MAPLE
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G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(1+t*z+(1-t)*sqrt(1-4*z)+t*sqrt(1-6*z+5*z^2))\ : Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 15 do seq(coeff(P[n], t, j), j=0..floor((n+1)/3)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A000108, A129158, A129154.
Sequence in context: A096976 A119245 A128731 this_sequence A086905 A167638 A054651
Adjacent sequences: A129154 A129155 A129156 this_sequence A129158 A129159 A129160
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2007
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