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1, 1, 2, 1, 2, 2, 4, 5, 10, 14, 28, 42, 84, 132, 264, 429, 858, 1430, 2860, 4862, 9724, 16796, 33592, 58786, 117572, 208012, 416024, 742900, 1485800, 2674440, 5348880, 9694845, 19389690, 35357670, 70715340, 129644790, 259289580, 477638700
(list;
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listen;
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internal format)
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OFFSET
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0,3
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COMMENTS
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The number of n-node binary trees fixed by the corresponding automorphism(s). Essentially A000108 interleaved with A068875.
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LINKS
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FORMULA
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G.f.: (1+4x-(1+2x)sqrt(1-4x^2))/(2x). - Paul Barry, Apr 11 2005
D-finite with recurrence: (n+1)*a(n) - 2*a(n-1) + 4(3-n)*a(n-2) = 0. - R. J. Mathar, Dec 17 2011, corrected by Georg Fischer, Feb 13 2020
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MAPLE
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seq(seq(binomial(2*j, j)/(1+j)*i, i=1..2), j=0..19); # Zerinvary Lajos, Apr 29 2007
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MATHEMATICA
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a[0] = 1; a[n_] := If[EvenQ[n], 2*CatalanNumber[n/2 - 1], CatalanNumber[(n-1)/2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 24 2013 *)
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PROG
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(Scheme) (define (A089408 n) (cond ((zero? n) 1) ((even? n) (* 2 (A000108 (-1+ (/ n 2))))) (else (A000108 (/ (-1+ n) 2)))))
(Python)
from sympy import catalan
def a(n): return 1 if n==0 else 2*catalan(n//2 - 1) if n%2==0 else catalan((n - 1)//2) # Indranil Ghosh, May 23 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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