A005354 - OEIS

%I M2808 #48 Jan 31 2018 10:52:38

%S 1,1,0,0,0,1,3,9,28,85,262,827,2651,8626,28507,95393,322938,1104525,

%T 3812367,13266366,46504495,164098390,582521687,2079133141,7457788295,

%U 26872946466,97238824018,353218128299,1287657977946,4709784136316

%N Number of asymmetric planar trees with n nodes.

%C a(13) in the Labelle table is a typographical error. - _R. J. Mathar_, Feb 03 2010

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A005354/b005354.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from Vincenzo Librandi)

%H Gilbert Labelle, <a href="http://dx.doi.org/10.1016/0747-7171(92)90037-5">Counting asymmetric enriched trees</a>, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.

%H Torsten Mütze and Franziska Weber, <a href="http://arxiv.org/abs/1111.2413">Construction of 2-factors in the middle layer of the discrete cube</a>, arXiv preprint arXiv:1111.2413 [math.CO], 2011.

%H T. Mütze and F. Weber, <a href="http://dx.doi.org/10.1016/j.jcta.2012.06.005">Construction of 2-factors in the middle layer of the discrete cube</a>, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.

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

%F From _Christian G. Bower_, Dec 15 1999: (Start)

%F G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A022553(n-1) and C is g.f. of A000108(n-1).

%F a(n) = A022553(n-1) - A000108(n-2)/2 - (if n is even) A000108(n/2-1)/2. (End)

%p From _R. J. Mathar_, Feb 03 2010: (Start)

%p A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:

%p A007727 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do a := a+binomial(2*d,d)*numtheory[mobius](n/d) ; end do ; a ; end proc;

%p A022553 := proc(n) A007727(n)/2/n ; end proc:

%p A005354 := proc(n) local a; if n <=1 then 1; else a := A022553(n-1) ; a := a-A000108(n-1)/2 ; if type(n,'even') then a := a-A000108(n/2-1)/2 ; end if; a ; end if; end proc: seq(A005354(n),n=0..20) ; (End)

%t a[0] = a[1] = 1; a[n_] := DivisorSum[n-1, MoebiusMu[(n-1)/#]*Binomial[2#, #]&]/(2(n-1)) - CatalanNumber[n-1]/2 - Boole[EvenQ[n]]*CatalanNumber[n/2 - 1]/2; Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, May 09 2012, after _R. J. Mathar_, updated Jan 31 2018 *)

%Y Cf. A000108, A002995, A022553.

%K nonn,nice

%O 0,7

%A _N. J. A. Sloane_, _Simon Plouffe_, _Susanna Cuyler_

%E More terms from _Christian G. Bower_, Dec 15 1999