cp's OEIS Frontend

A089849 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A069772.

%I A089849 #45 Feb 22 2025 09:01:45
%S A089849 1,1,2,1,6,2,20,5,70,14,252,42,924,132,3432,429,12870,1430,48620,4862,
%T A089849 184756,16796,705432,58786,2704156,208012,10400600,742900,40116600,
%U A089849 2674440,155117520,9694845,601080390,35357670,2333606220,129644790
%N A089849 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A069772.
%C A089849 The number of n-node binary trees fixed by the corresponding automorphism(s). A000984 interleaved with A000108.
%H A089849 G. C. Greubel, <a href="/A089849/b089849.txt">Table of n, a(n) for n = 0..1000</a>
%H A089849 Antti Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing the initial terms of this sequence</a>.
%F A089849 a(2n) = A000984(n), a(2n+1) = A000108(n).
%F A089849 a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,n-k)*Catalan(k). - _Paul Barry_, Feb 23 2005
%F A089849 From _Paul Barry_, Jan 23 2006: (Start)
%F A089849 a(n+1) = Jacobi_P(n, 2, 0, 0)*2^n*(cos(Pi*n/2)+sin(Pi*n/2)).
%F A089849 a(n+1) = (Sum_{k=0..n} C(n,k)*C(n+2,k)*(-1)^k)*(cos(Pi*n/2)+sin(Pi*n/2)). (End)
%F A089849 From _Sergei N. Gladkovskii_, Dec 18 2012 (Start)
%F A089849 E.g.f.: 1 + integral(G(0)) dx where G(k) = 1 + 2*x/(1 - 2*x/(2*x + (2*k+2)*(2*k+4)/G(k+1) )); (continued fraction).
%F A089849 E.g.f.: 1 + x*G(0) where G(k) = 1 + x*(2*k+1)/(k+1 - x*(k+1)/(x + (k+2)*(2*k+3)/G(k+1) )); (continued fraction).
%F A089849 E.g.f.: E(x) = integral( (1/x + 2)*BesselI(1,2*x) ) dx. (End)
%F A089849 G.f.: G(0), where G(k) = 1 + x/(k+1 - (k+1)*(4*k+2)*x/((4*k+2)*x + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 19 2013
%F A089849 From _Amiram Eldar_, Mar 12 2023: (Start)
%F A089849 Sum_{n>=0} 1/a(n) = 10/3 + 2*Pi/(3*sqrt(3)).
%F A089849 Sum_{n>=0} (-1)^n/a(n) = 2/3 + 2*Pi/(9*sqrt(3)). (End)
%t A089849 a[n_] := If[EvenQ[n], Binomial[n, n/2], CatalanNumber[(n-1)/2]];
%t A089849 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Mar 09 2023 *)
%o A089849 (Scheme) (define (A089849 n) (if (even? n) (A000984 (/ n 2)) (A000108 (/ (- n 1) 2))))
%o A089849 (Magma)
%o A089849 A089849:= func< n | n eq 0 select 1 else ((1+(-1)^n)*Binomial(n,Floor(n/2))+(1-(-1)^n)*Catalan(Floor((n-1)/2)))/2 >;
%o A089849 [A089849(n): n in [0..50]]; // _G. C. Greubel_, Feb 22 2025
%o A089849 (SageMath)
%o A089849 def A089849(n): return binomial(n, n//2) if n%2==0 else catalan_number((n-1)//2)
%o A089849 print([A089849(n) for n in range(51)]) # _G. C. Greubel_, Feb 22 2025
%Y A089849 Cf. A014137, A014138, A069772, A089880.
%Y A089849 Cf. A000984 interleaved with A000108.
%K A089849 nonn,easy
%O A089849 0,3
%A A089849 _Antti Karttunen_, Nov 29 2003