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A005354 Number of asymmetric planar trees with n nodes.
(Formerly M2808)
4
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 (list; graph; refs; listen; history; text; internal format)
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.
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
Sequence in context: A033139 A291731 A291257 * A084084 A091140 A052541
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Christian G. Bower, Dec 15 1999
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)