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Search: id:A101371
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| A101371 |
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves at level 1. |
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+0
1
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| 1, 0, 1, 2, 0, 1, 7, 4, 0, 1, 34, 14, 6, 0, 1, 171, 72, 21, 8, 0, 1, 905, 370, 114, 28, 10, 0, 1, 4952, 1995, 597, 160, 35, 12, 0, 1, 27802, 11064, 3278, 852, 210, 42, 14, 0, 1, 159254, 62774, 18420, 4762, 1135, 264, 49, 16, 0, 1, 927081, 362614, 105618, 27104, 6455, 1446, 322, 56, 18, 0, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row n has n+1 terms. Row sums give A001764. Column 0 gives A023053.
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REFERENCES
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P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
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FORMULA
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T(n, k)=sum((-1)^i*[(k+i+1)/(2n-k+1)]binomial(k+i, i)binomial(3n-2k-2i, n-k-i), i=0..n-k) (0<=k<=n). G.f.=g/(1+zg-tzg), where g=1+zg^3.
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EXAMPLE
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Triangle begins:
1;
0,1;
2,0,1;
7,4,0,1;
34,14,6,0,1;
171,72,21,8,0,1;
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MAPLE
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T:=proc(n, k) if k<=n then sum((-1)^i*(k+i+1)*binomial(k+i, i)*binomial(3*n-2*k-2*i, n-k-i)/(2*n-k-i+1), i=0..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, A023053.
Sequence in context: A138106 A131689 A114329 this_sequence A078341 A065329 A108998
Adjacent sequences: A101368 A101369 A101370 this_sequence A101372 A101373 A101374
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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 14 2005
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