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Search: id:A127537
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| A127537 |
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Triangle read by rows: T(n,k) (n >=2, 1<=k <=2n-3) is the number of non-crossing connected graphs on n nodes on a circle, having k edges. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,.... |
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+0
1
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| 1, 0, 3, 1, 0, 0, 12, 9, 2, 0, 0, 0, 55, 66, 30, 5, 0, 0, 0, 0, 273, 455, 315, 105, 14, 0, 0, 0, 0, 0, 1428, 3060, 2856, 1428, 378, 42, 0, 0, 0, 0, 0, 0, 7752, 20349, 23940, 15960, 6300, 1386, 132, 0, 0, 0, 0, 0, 0, 0, 43263, 134596, 191268, 159390, 83490, 27324, 5148, 429
(list; table; graph; listen)
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OFFSET
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2,3
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COMMENT
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Row n contains 2n-3 terms, the first n-2 of which are equal to 0. T(n,n-1)=A001764(n-1). T(n,2n-3)=A000108(n-2) (the Catalan numbers). T(n,k)=A089434(n,k+1-n) Sum(k*T(n,k),k=n-1..2n-3)=A045741(n). Sum(T(n,k),n=k..2k-2)=A065065(k).
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REFERENCES
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C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
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)=C(3n-3,n+k)C(k-1,k-n+1)/(n-1) (n>=2, 0<=k<=2n-3). G.f.=G=G(t,z) satisfies tG^3+tG^2-z(1+2t)G+z^2*(1+t)=0.
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EXAMPLE
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Triangle starts:
1;
0,3,1;
0,0,12,9,2;
0,0,0,55,66,30,5;
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MAPLE
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T:=(n, k)->binomial(3*n-3, n+k)*binomial(k-1, k-n+1)/(n-1): for n from 2 to 10 do seq(T(n, k), k=1..2*n-3) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001764, A000108, A089434, A045741, A065065.
Sequence in context: A091925 A034370 A144402 this_sequence A025443 A120080 A111700
Adjacent sequences: A127534 A127535 A127536 this_sequence A127538 A127539 A127540
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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 24 2007
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