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Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089864.
90

%I #33 Jun 06 2020 15:32:39

%S 1,1,2,1,2,2,4,5,10,14,28,42,84,132,264,429,858,1430,2860,4862,9724,

%T 16796,33592,58786,117572,208012,416024,742900,1485800,2674440,

%U 5348880,9694845,19389690,35357670,70715340,129644790,259289580,477638700

%N Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089864.

%C The number of n-node binary trees fixed by the corresponding automorphism(s). Essentially A000108 interleaved with A068875.

%H Indranil Ghosh, <a href="/A089408/b089408.txt">Table of n, a(n) for n = 0..1000</a>

%H Antti Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing the initial terms of this sequence</a>

%F a(0)=1, a(2n) = 2*A000108(n-1), a(2n+1) = A000108(n)

%F G.f.: (1+4x-(1+2x)sqrt(1-4x^2))/(2x). - _Paul Barry_, Apr 11 2005

%F C(2*j,j)/(1+j)*i, i=1..2), j >= 0. - _Zerinvary Lajos_, Apr 29 2007

%F 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

%p seq(seq(binomial(2*j,j)/(1+j)*i, i=1..2),j=0..19); # _Zerinvary Lajos_, Apr 29 2007

%t 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 *)

%o (Scheme) (define (A089408 n) (cond ((zero? n) 1) ((even? n) (* 2 (A000108 (-1+ (/ n 2))))) (else (A000108 (/ (-1+ n) 2)))))

%o (Python)

%o from sympy import catalan

%o 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

%Y Cf. A089402.

%Y Cf. A000108.

%K nonn,easy

%O 0,3

%A _Antti Karttunen_, Nov 29 2003