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Search: id:A102593
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| A102593 |
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the maximum number of contiguous border edges starting from the root in counterclockwise direction is equal to k. |
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+0
2
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| 1, 0, 1, 1, 1, 1, 5, 4, 2, 1, 25, 18, 8, 3, 1, 130, 88, 37, 13, 4, 1, 700, 455, 185, 63, 19, 5, 1, 3876, 2448, 973, 325, 97, 26, 6, 1, 21945, 13566, 5304, 1748, 518, 140, 34, 7, 1, 126500, 76912, 29697, 9690, 2856, 775, 193, 43, 8, 1, 740025, 444015, 169763, 54967
(list; table; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row n has n+1 terms. Row sums yield the ternary numbers (A001764). T(n,0)=A102893(n).
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REFERENCES
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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
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T(n, k)=(k+1)binomial(3n-2k, n-k)/(2n-k+1)-(k+2)binomial(3n-2k-2, n-k-1)/(2n-k) if n>1, 0<=k<=n; T(1, 1)=1; T(0, 0)=1; T(n, k)=0 if k>n. G.f.=G(t, z)=g(1-zg)/(1-tzg), where g=1+zg^3 is the g.f. for the ternary numbers (A001764).
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EXAMPLE
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T(2,0)=T(2,1)=T(2,2)=1 because in _\, /\, and /_ the maximum number of contiguous border edges starting from the root in counterclockwise direction is 0,1, and 2, respectively.
Triangle starts:
1;
0,1;
1,1,1;
5,4,2,1;
25,18,8,3,1;
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MAPLE
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T:=proc(n, k) if n=0 and k=0 then 1 elif n=1 and k=1 then 1 elif k<=n then (k+1)*binomial(3*n-2*k, n-k)/(2*n-k+1)-(k+2)*binomial(3*n-2*k-2, n-k-1)/(2*n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001764, A102893.
Adjacent sequences: A102590 A102591 A102592 this_sequence A102594 A102595 A102596
Sequence in context: A102220 A109430 A085917 this_sequence A090462 A081749 A074825
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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), Jan 22 2005
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