A054357 - OEIS

%I #51 Jan 09 2024 10:54:18

%S 1,1,2,3,6,10,28,63,190,546,1708,5346,17428,57148,191280,646363,

%T 2210670,7626166,26538292,93013854,328215300,1165060668,4158330416,

%U 14915635378,53746119972,194477856100,706437056648,2575316704200,9419571138368

%N Number of unlabeled 2-ary cacti having n polygons. Also number of bicolored plane trees with n edges.

%C a(n) = the number of inequivalent non-crossing partitions of n points (equally spaced) on a circle, under rotations of the circle. This may be considered the number of non-crossing partitions of n unlabeled points on a circle, so this sequence has the same relation to the Catalan numbers (A000108) as the number of partitions of an integer (A000041) has to the Bell numbers (A000110). - _Len Smiley_, Sep 06 2005

%H Indranil Ghosh, <a href="/A054357/b054357.txt">Table of n, a(n) for n = 0..1000</a>

%H Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, <a href="https://doi.org/10.1006/aama.1999.0665">Enumeration of m-ary cacti</a>, Advances in Applied Mathematics, 24 (2000), 22-56.

%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rot">Partition related number triangles</a>

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>

%F a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(2*d, d)) - binomial(2*n, n)/(n+1) for n > 0. - _Andrew Howroyd_, May 02 2018

%F a(n) ~ 2^(2*n) / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Jul 17 2017

%t a[n_] := If[n == 0, 1, (Binomial[2*n, n]/(n + 1) + DivisorSum[n, Binomial[2*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Jul 17 2017 *)

%o (PARI) a(n)=if(n==0, 1, (binomial(2*n, n)/(n + 1) + sumdiv(n, d, binomial(2*d, d)*eulerphi(n/d)*(d<n)))/n); \\ _Indranil Ghosh_, Jul 17 2017

%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1)) \\ _Andrew Howroyd_, May 02 2018

%o (Python)

%o from sympy import binomial, divisors, totient

%o def a(n): return 1 if n==0 else (binomial(2*n, n)//(n + 1) + sum(binomial(2*d, d)*totient(n//d)*(d<n) for d in divisors(n)))//n

%o print([a(n) for n in range(31)]) # _Indranil Ghosh_, Jul 17 2017

%Y Column k=2 of A303912.

%Y Row sums of A209805.

%Y Cf. A002995, A054358, A111275.

%K nonn

%O 0,3

%A _Simon Plouffe_

%E More terms from _Len Smiley_, Sep 06 2005

%E More terms from _Vladeta Jovovic_, Oct 04 2007