A005354 Number of asymmetric planar trees with n nodes. (Formerly M2808)

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

0,7

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

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)

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 *)

Crossrefs

Cf. A000108, A002995, A022553.

Sequence in context: A372870 A291731 A291257 * A084084 A091140 A052541

Adjacent sequences: A005351 A005352 A005353 * A005355 A005356 A005357

Keyword

nonn,nice

Author

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

Extensions

More terms from Christian G. Bower, Dec 15 1999

Status

approved