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Search: id:A143020
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| A143020 |
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Sum of the distances from a fixed node (root) to the next node in all non-crossing graphs on n nodes on a circle. |
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+0
2
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| 1, 5, 31, 218, 1658, 13293, 110675, 947870, 8297926, 73924162, 668038006, 6108962580, 56426393268, 525673683069, 4933634156571, 46604425575734, 442753710351950, 4227598589181750, 40549714320544770, 390522305786747820
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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a(n)=Sum(A143018(n,k),k=1..n-1).
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REFERENCES
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P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discr. Math. 204 (1999), 203-229.
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FORMULA
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a(n)=Sum(k*T(n,k), k=1..n-1), where T(n,k)=k*L(n-k-1,3n-k-4,n-1)/(n-1) (n>=2, 1<=k<=n--1), with L(p,q,r)=[u^p](1+u)^q/(1-u)^r = Sum[binom(q,i)binom(r+p-1-i,r-1), i=0..min(p,q)] (T(n,k) is A143018). G.f. = g(g-z)/z, where g=g(z), the g.f. for the number of non-crossing connected graphs on n nodes on a circle, satisfies g^3 + g^2 -3zg +2z^2=0 (A007297).
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EXAMPLE
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a(3)=5 because in the graphs (AB,BC,CA), (AB,AC), (AB,BC), and (AC,BC) the distances from A to B are 1, 1, 1, and 2, respectively.
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MAPLE
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L:=proc(p, q, r) options operator, arrow: sum(binomial(q, i)*binomial(r+p-1-i, r-1), i=0..min(p, q)) end proc: T:=proc(n, k) options operator, arrow: k*L(n-k-1, 3*n-k-4, n-1)/(n-1) end proc: seq(add(k*T(n, k), k=1..n-1), n=2..22);
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CROSSREFS
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Cf. A007297, A143018.
Sequence in context: A036758 A110379 A097146 this_sequence A059035 A058309 A001910
Adjacent sequences: A143017 A143018 A143019 this_sequence A143021 A143022 A143023
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 30 2008
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