A130380 -id:A130380 - cp's OEIS frontend

A130918 Simple self-inverse permutation of natural numbers: List each block of A000108(n) numbers from A014137(n-1) to A014138(n-1) in reverse order.

0, 1, 3, 2, 8, 7, 6, 5, 4, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 196, 195, 194, 193, 192

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

In principle this involution is the signature permutation of yet another Catalan automorphism. However, the question remains what is the most "natural" way to create such an automorphism acting e.g. on S-expressions (i.e. rooted plane binary trees), which would produce this sequence as its signature permutation.

A. Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Signature permutations of Catalan bijections
Index entries for signature-permutations of Catalan automorphisms

Inverse: A130918. Cf. A054429, A057163. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A130380 and A036987.

(define (A130918 n) (if (zero? n) n (- (A014138 (- (A072643 n) 1)) (A082853 n))))

a(0)=0, a(n) = A014138(A072643(n)-1) - A082853(n).

A028304 a(n) = ceiling( binomial(n, floor(n/2))/(1 + ceiling(n/2)) ) (interpolates between Catalan numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 14, 21, 42, 66, 132, 215, 429, 715, 1430, 2431, 4862, 8398, 16796, 29393, 58786, 104006, 208012, 371450, 742900, 1337220, 2674440, 4847423, 9694845, 17678835, 35357670, 64822395, 129644790, 238819350, 477638700, 883631595, 1767263190, 3282060210

Offset: 0

N. J. A. Sloane

nonn

D. Miklos et al., eds., Combinatorics, Paul Erdős is Eighty, Bolyai Math. Soc., 1993, Vol. 1, p. 101.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A000245, A001405, A004526, A130380.

[Ceiling(Binomial(n,Floor(n/2))/Floor((n+3)/2)): n in [0..50]]; // G. C. Greubel, Jan 05 2024

A028304 := proc(n)
    A001405(n)/(ceil(n/2)+1) ;
    ceil(%) ;
end proc: # R. J. Mathar, Dec 15 2015

Table[Ceiling[(1/(Ceiling[n/2] + 1)) Binomial[n, Floor[n/2]]], {n, 0, 49}] (* Alonso del Arte, Oct 30 2019 *)

[ceil(binomial(n,int(n/2))/((n+3)//2)) for n in range(51)] # G. C. Greubel, Jan 05 2024

a(2*n) = A000108(n), a(2*n+1) = A130380(n+1). - R. J. Mathar, Dec 15 2015

a(n) = ceiling(A001405(n)/A004526(n+3)). - G. C. Greubel, Jan 05 2024

cp's OEIS Frontend

A130918 Simple self-inverse permutation of natural numbers: List each block of A000108(n) numbers from A014137(n-1) to A014138(n-1) in reverse order.

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A028304 a(n) = ceiling( binomial(n, floor(n/2))/(1 + ceiling(n/2)) ) (interpolates between Catalan numbers).

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