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A005354
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Number of asymmetric planar trees with n nodes.
(Formerly M2808)
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4
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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
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OFFSET
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0,7
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COMMENTS
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a(13) in the Labelle table is a typographical error. - R. J. Mathar, Feb 03 2010
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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).
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MAPLE
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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;
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)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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