A089849 - OEIS

A089849

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

1, 1, 2, 1, 6, 2, 20, 5, 70, 14, 252, 42, 924, 132, 3432, 429, 12870, 1430, 48620, 4862, 184756, 16796, 705432, 58786, 2704156, 208012, 10400600, 742900, 40116600, 2674440, 155117520, 9694845, 601080390, 35357670, 2333606220, 129644790

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OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..35.

Antti Karttunen, C-program for computing the initial terms of this sequence.

FORMULA

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

a(n) = Sum_{k=0..floor(n/2)} C(k)*C(k+1,n-k). - Paul Barry, Feb 23 2005

From Paul Barry, Jan 23 2006: (Start)

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

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)

From Sergei N. Gladkovskii, Dec 18 2012 (Start)

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).

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).

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

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

From Amiram Eldar, Mar 12 2023: (Start)

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

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

MATHEMATICA

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

Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 09 2023 *)

PROG

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

CROSSREFS

Cf. A089880, A014137, A014138, A069772.

Cf. A000984 interleaved with A000108.

Sequence in context: A057560 A085592 A174421 * A185330 A217955 A325703

Adjacent sequences: A089846 A089847 A089848 * A089850 A089851 A089852

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Nov 29 2003

STATUS

approved