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Search: id:A101449
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| A101449 |
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Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k nonroot nodes of degree 1. |
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+0
2
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| 1, 1, 2, 4, 4, 4, 11, 24, 12, 8, 41, 88, 96, 32, 16, 146, 410, 440, 320, 80, 32, 564, 1752, 2460, 1760, 960, 192, 64, 2199, 7896, 12264, 11480, 6160, 2688, 448, 128, 8835, 35184, 63168, 65408, 45920, 19712, 7168, 1024, 256, 35989, 159030, 316656, 379008
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row n contains n terms. Row sums yield the ternary numbers (A001764). Column 0 is A030981. T(n,k)=2^k*binomial(n-1,k)*A030981(n-k). The average number of nonroot nodes of degree 1 over all noncrossing trees with n edges is 4n(n-1)(2n+1)/[3(3n-1)(3n-2)] ~ 8n/27.
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REFERENCES
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P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math. 180, 1998, 301-313.
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FORMULA
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T(n, k)=[2^k/(n-k)]*binomial(n-1, k)*sum((-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3i, i-1), i=1..n-k) (0<=k<n).
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EXAMPLE
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T(2,0)=1 (/\); T(2,1)=2 (/_, _\ ).
Triangle begins:
1;
1,2;
4,4,4;
11,24,12,8;
41,88,96,32,16;
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MAPLE
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T:=proc(n, k) if k<n then 2^k*binomial(n-1, k)*sum((-1)^(n-k-i)*2^(n-k-i)*binomial(n-k, i)*binomial(3*i, i-1), i=1..n-k)/(n-k) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; #yields sequence in triangular form
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CROSSREFS
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Cf. A001764, A030981.
Sequence in context: A021413 A082855 A107058 this_sequence A134188 A140295 A070529
Adjacent sequences: A101446 A101447 A101448 this_sequence A101450 A101451 A101452
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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 17 2005
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