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A089073 Number of symmetric non-crossing connected graphs on n equidistant nodes on a circle. 0
1, 1, 2, 5, 10, 32, 64, 231, 462, 1792, 3584, 14586, 29172, 122880, 245760, 1062347, 2124694, 9371648, 18743296, 84021990, 168043980, 763363328, 1526726656, 7012604550, 14025209100, 65028489216, 130056978432, 607892634420 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Number of symmetric non-crossing connected graphs on n equidistant nodes on a circle (it is assumed that the axis of symmetry is a diameter of the circle passing through a given node). Example: a(4)=5 because on the nodes A,B,C,D (axis of symmetry through A) the only symmetric non-crossing connected graphs are {AB,AC,AD), (AC,BC,DC), (AB,BC,CD,DA), (AB,BC,CD,DA,BD), (AB,BC,CD,DA,AC).

LINKS

Table of n, a(n) for n=1..28.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.

FORMULA

a(2k) = 4^k*binomial((3k-1)/2, k)/[2(k+1)], a(2k+1) = 2a(2k).

a(2k) = (1/2)A078531(k), a(2k+1) = A078531(k).

MAPLE

a := proc(n) if n mod 2 = 0 then 4^(n/2)*binomial((3*(n/2)-1)/2, n/2)/2/(n/2+1) else 2*4^((n-1)/2)*binomial((3*((n-1)/2)-1)/2, (n-1)/2)/2/((n-1)/2+1) fi end; seq(a(n), n=1..30);

MATHEMATICA

a[n_] := If[EvenQ[n], 2^n Binomial[(3n-2)/4, n/2]/(n+2), 2^n Binomial[ (3n-5)/4, (n-1)/2]/(n+1)];

Array[a, 28] (* Jean-Fran├žois Alcover, Jul 29 2018 *)

CROSSREFS

Cf. A078531.

Sequence in context: A226963 A018386 A270521 * A343167 A138190 A056300

Adjacent sequences:  A089070 A089071 A089072 * A089074 A089075 A089076

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 04 2003

EXTENSIONS

Name edited by Michel Marcus, Jul 30 2018

STATUS

approved

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Last modified July 24 06:04 EDT 2021. Contains 346273 sequences. (Running on oeis4.)