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Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A069772.
5

%I #38 Jun 17 2024 07:10:34

%S 1,1,2,1,6,2,20,5,70,14,252,42,924,132,3432,429,12870,1430,48620,4862,

%T 184756,16796,705432,58786,2704156,208012,10400600,742900,40116600,

%U 2674440,155117520,9694845,601080390,35357670,2333606220,129644790

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

%C The number of n-node binary trees fixed by the corresponding automorphism(s). A000984 interleaved with A000108.

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

%F a(2n) = A000984(n), a(2n+1) = A000108(n).

%F a(n) = Sum_{k=0..floor(n/2)} C(k)*C(k+1,n-k). - _Paul Barry_, Feb 23 2005

%F From _Paul Barry_, Jan 23 2006: (Start)

%F a(n+1) = Jacobi_P(n, 2, 0, 0)*2^n*(cos(Pi*n/2)+sin(Pi*n/2)).

%F 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 From _Sergei N. Gladkovskii_, Dec 18 2012 (Start)

%F 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) )); (recursively defined continued fraction).

%F 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) )); (recursively defined continued fraction).

%F E.g.f.: E(x) = integral( (1/x + 2)*BesselI(1,2*x) ) dx. (End)

%F 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 From _Amiram Eldar_, Mar 12 2023: (Start)

%F Sum_{n>=0} 1/a(n) = 10/3 + 2*Pi/(3*sqrt(3)).

%F Sum_{n>=0} (-1)^n/a(n) = 2/3 + 2*Pi/(9*sqrt(3)). (End)

%t a[n_] := If[EvenQ[n], Binomial[n, n/2], CatalanNumber[(n-1)/2]];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Mar 09 2023 *)

%o (Scheme) (define (A089849 n) (if (even? n) (A000984 (/ n 2)) (A000108 (/ (- n 1) 2))))

%Y Cf. A089880, A014137, A014138, A069772.

%Y Cf. A000984 interleaved with A000108.

%K nonn,easy

%O 0,3

%A _Antti Karttunen_, Nov 29 2003