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A102595
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the maximal number of contiguous border edges starting from the root in both directions is equal to k.
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0
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1, 0, 1, 0, 0, 3, 1, 4, 3, 4, 7, 20, 15, 8, 5, 42, 102, 72, 36, 15, 6, 245, 540, 366, 176, 70, 24, 7, 1428, 2950, 1944, 912, 355, 120, 35, 8, 8379, 16524, 10668, 4920, 1890, 636, 189, 48, 9, 49588, 94430, 60021, 27336, 10405, 3492, 1050, 280, 63, 10, 296010
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row n has n+1 terms. Row sums yield the ternary numbers (A001764). T(n,0)=A102594(n).
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REFERENCES
| M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math.180, 1998, 301-313.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
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FORMULA
| G.f.=G(t, z)=(g+zg-tz-2zg^2+t^2*(1-t)z^3*g^2-2t(1-t)z^2*g)/(1-tzg)^2, where g=1+zg^3 is the g.f. for the ternary numbers (A001764).
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EXAMPLE
| T(2,0)=T(2,1)=0, T(2,2)=3 because in all the noncrossing trees _\, /\ and /_, the maximal number of contiguous border edges starting
from the root in both directions is equal to 2.
Triangle starts:
1;
0,1;
0,0,3;
1,4,3,4;
7,20,15,8,5;
42,102,72,36,15,6;
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MAPLE
| G:=(g+z*g-t*z-2*z*g^2+t^2*(1-t)*z^3*g^2-2*t*(1-t)*z^2*g)/(1-t*z*g)^2: z:=w^2: b:=w*sqrt(3): g:=2*sin(arcsin(3*b/2)/3)/b: Gser:=simplify(series(G, w=0, 24)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gser, w^(2*n))) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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MATHEMATICA
| max = 20; z = w^2; b = w*Sqrt[3]; g = 2*(Sin[ ArcSin[3*(b/2)]/3]/b); gf = (g + z*g - t*z - 2*z*g^2 + t^2*(1 - t)*z^3*g^2 - 2*t*(1 - t)*z^2*g)/(1 - t*z*g)^2; se = Series[gf, {w, 0, max}]; Flatten[ Rest /@ DeleteCases[ (CoefficientList[t*#1, t] & ) /@ CoefficientList[se, w], {}]] (* From Jean-François Alcover, Oct 05 2011, after Maple *)
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CROSSREFS
| Cf. A001764, A102594.
Sequence in context: A152842 A082909 A029151 * A113415 A054019 A035626
Adjacent sequences: A102592 A102593 A102594 * A102596 A102597 A102598
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 22 2005
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