A005354 Number of asymmetric planar trees with n nodes.

1, 1, 0, 0, 0, 1, 3, 9, 28, 85, 262, 827, 2651, 8626, 28507, 95393, 322938, 1104525, 3812367, 13266366, 46504495, 164098390, 582521687, 2079133141, 7457788295, 26872946466, 97238824018, 353218128299, 1287657977946, 4709784136316

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

Keywords

• nonn
• nice

Comments

a(13) in the Labelle table is a typographical error. - R. J. Mathar, Feb 03 2010

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
• Gilbert Labelle, Counting asymmetric enriched trees, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.
• Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
• T. Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.
• Index entries for sequences related to trees

Crossrefs

Cf. A000108, A002995, A022553.

Programs

Maple

From R. J. Mathar, Feb 03 2010: (Start)
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
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;
A022553 := proc(n) A007727(n)/2/n ; end proc:
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)

Mathematica

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 *)

Formula

From Christian G. Bower, Dec 15 1999: (Start)

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).

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

Extensions

More terms from Christian G. Bower, Dec 15 1999