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A143024
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Triangle read by rows: T(n,k) is the number of non-crossing connected graphs on n nodes on a circle having root (a distinguished node) of degree 1 and having k edges (n>=2, 1<=k<=2n-4).
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0
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1, 0, 2, 0, 0, 7, 2, 0, 0, 0, 30, 20, 4, 0, 0, 0, 0, 143, 156, 65, 10, 0, 0, 0, 0, 0, 728, 1120, 720, 224, 28, 0, 0, 0, 0, 0, 0, 3876, 7752, 6783, 3192, 798, 84, 0, 0, 0, 0, 0, 0, 0, 21318, 52668, 58520, 36960, 13860, 2904, 264, 0, 0, 0, 0, 0, 0, 0, 0, 120175, 354200, 478170
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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COMMENTS
| Row n contains 2n-4 terms, the first n-2 of which are 0.
Row sums yield A089436.
T(n,n-1)=A006013(n-2)..
Sum(k*T(n,k),k=2..2n-4)=A143025.
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REFERENCES
| C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discr. Math. 204 (1999), 203-229.
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FORMULA
| T(n,k)=2*binom(k-2, n-3)binom(3n-5, 2n-k-4)/(n-2) (n>=3, 2<=k<=2n-4); T(2,1)=1; T(2,k)=0 (k>=2). The trivariate g.f. G=G(t,s,z) for non-crossing connected graphs on nodes on a circle, with respect to number of nodes (marked by z), number of edges (marked by t) and degree of root (marked by s) is G=z + tszg^2/[z-ts(g - z + g^2)], where g=g(t,z) satisfies tg^3 + tg^2 - (1 + 2t)zg +(1 + t)z^2 = 0 (see Domb & Barrett, Eq. (47); Flajolet & Noy, Eq. (18)).
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EXAMPLE
| T(3,2)=2 because we have {AB,BC} and {AC, BC} (A is the root).
Triangle starts:
1;
0,2;
0,0,7,2;
0,0,0,30,20,4;
0,0,0,0,143,156,65,10;
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MAPLE
| T:=proc(n, k) options operator, arrow: 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4)/(n-2) end proc: 1; for n from 3 to 10 do 0, seq(T(n, k), k=2..2*n-4) end do; % yields sequence in triangular form
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CROSSREFS
| Cf. A007297, A089436, A006013, A143025.
Sequence in context: A161800 A100344 A094596 * A198232 A160213 A192058
Adjacent sequences: A143021 A143022 A143023 * A143025 A143026 A143027
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 31 2008
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