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A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges. 5
1, 2, 7, 33, 181, 1083, 6854, 45111, 305629, 2117283, 14929212, 106790500, 773035602, 5652275723, 41683912721, 309691336359, 2315772552485, 17415395593371, 131632335068744, 999423449413828, 7618960581522348, 58295017292748756, 447517868947619432, 3445923223190363608 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.

LINKS

Table of n, a(n) for n=1..24.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).

Source

Philippe Flajolet, Enumeration of planar configurations in computational geometry

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486, 502

FORMULA

a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).

G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). [Vladimir Kruchinin, Nov 25 2011]

From Peter Bala, Nov 07 2015: (Start)

O.g.f. A(x) = revert(x/((1 + x)*C(x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.

Row sums of A094021. (End)

MAPLE

ZZ:=[F, {F=Union(Epsilon, ZB), ZB=Prod(Z1, P), P=Sequence(B), B=Prod(P, Z1, P), Z1=Prod(Z, F)}, unlabeled]: seq(count(ZZ, size=n), n=1..20); # Zerinvary Lajos, Apr 22 2007

MATHEMATICA

a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* Jean-Fran├žois Alcover, Sep 05 2012, after formula *)

PROG

(PARI) N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 25 2016

CROSSREFS

Cf. A006013, A000108, A094021.

Sequence in context: A162257 A214954 A055724 * A086618 A224769 A249636

Adjacent sequences:  A054724 A054725 A054726 * A054728 A054729 A054730

KEYWORD

nonn,easy

AUTHOR

Philippe Flajolet, Apr 20 2000

STATUS

approved

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Last modified December 8 19:05 EST 2016. Contains 278948 sequences.