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A089408 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089864.

%I A089408 #40 Jan 21 2025 09:27:24
%S A089408 1,1,2,1,2,2,4,5,10,14,28,42,84,132,264,429,858,1430,2860,4862,9724,
%T A089408 16796,33592,58786,117572,208012,416024,742900,1485800,2674440,
%U A089408 5348880,9694845,19389690,35357670,70715340,129644790,259289580,477638700
%N A089408 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089864.
%C A089408 The number of n-node binary trees fixed by the corresponding automorphism(s). Essentially A000108 interleaved with A068875.
%H A089408 Indranil Ghosh, <a href="/A089408/b089408.txt">Table of n, a(n) for n = 0..1000</a>
%H A089408 Antti Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing the initial terms of this sequence</a>
%H A089408 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 100.
%F A089408 a(0)=1, a(2n) = 2*A000108(n-1), a(2n+1) = A000108(n)
%F A089408 G.f.: (1+4x-(1+2x)sqrt(1-4x^2))/(2x). - _Paul Barry_, Apr 11 2005
%F A089408 a(2*j+i) = i*C(2*j,j)/(1+j), i=1..2, j >= 0. - _Zerinvary Lajos_, Apr 29 2007
%F A089408 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 A089408 seq(seq(binomial(2*j,j)/(1+j)*i, i=1..2),j=0..19); # _Zerinvary Lajos_, Apr 29 2007
%t A089408 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 A089408 (Scheme) (define (A089408 n) (cond ((zero? n) 1) ((even? n) (* 2 (A000108 (-1+ (/ n 2))))) (else (A000108 (/ (-1+ n) 2)))))
%o A089408 (Python)
%o A089408 from sympy import catalan
%o A089408 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 A089408 Cf. A089402.
%Y A089408 Cf. A000108.
%K A089408 nonn,easy
%O A089408 0,3
%A A089408 _Antti Karttunen_, Nov 29 2003