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Search: id:A101372
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| A101372 |
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Triangle read by rows: T(n,k) is number of leaves at level k in all noncrossing rooted trees on n+1 nodes. |
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1
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| 1, 2, 2, 7, 10, 4, 30, 50, 32, 8, 143, 260, 208, 88, 16, 728, 1400, 1280, 704, 224, 32, 3876, 7752, 7752, 5016, 2128, 544, 64, 21318, 43890, 46816, 33880, 17248, 5984, 1280, 128, 120175, 253000, 283360, 222640, 128800, 54400, 16000, 2944, 256
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row n has n terms. Row sums yield A045721. Column 1 is A006013.
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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)=2^(k-1)*[(3k-1)/(2n+k-1)]binomial(3n-2, n-k) (1<=k<=n). G.f.=tzg^2/(1-2tzg^3), where g=1+zg^3 is the g.f. of the ternary numbers (A001764).
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EXAMPLE
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Triangle begins:
1;
2,2;
7,10,4;
30,50,32,8;
143,260,208,88,16;
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MAPLE
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T:=(n, k)->2^(k-1)*(3*k-1)*binomial(3*n-2, n-k)/(2*n+k-1): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields triangle in triangular form
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CROSSREFS
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Cf. A045721, A006013.
Sequence in context: A070910 A107386 A095021 this_sequence A133374 A054226 A000024
Adjacent sequences: A101369 A101370 A101371 this_sequence A101373 A101374 A101375
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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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