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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 Cf. A000108, A002995, A022553. Sequence in context: A033139 A291731 A291257 * A084084 A091140 A052541 Adjacent sequences:  A005351 A005352 A005353 * A005355 A005356 A005357 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Christian G. Bower, Dec 15 1999 STATUS approved

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Last modified September 22 16:56 EDT 2019. Contains 327311 sequences. (Running on oeis4.)