A268082 -id:A268082 - cp's OEIS frontend

A201059 Denominator of binomial(2n,n)/(2n).

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 6, 13, 7, 1, 16, 17, 3, 19, 2, 7, 11, 23, 4, 25, 13, 27, 1, 29, 15, 31, 32, 11, 17, 5, 18, 37, 19, 39, 4, 41, 1, 43, 11, 1, 23, 47, 8, 49, 25, 17, 13, 53, 9, 55, 14, 19, 29, 59, 5, 61, 31, 21, 64, 13, 1, 67, 34, 23, 7, 71, 4, 73, 37, 5

Offset: 1

N. J. A. Sloane, Nov 26 2011

There is at least one published paper that refers to binomial(2n,n)/(2n) as the Catalan numbers. Of course the Catalan numbers are really A000108.

			1, 3/2, 10/3, 35/4, 126/5, 77, 1716/7, 6435/8, 24310/9, 46189/5, 352716/11, 676039/6, ...

David A. Corneth, Table of n, a(n) for n = 1..10000
Patrick Dehornoy, On the rotation distance between binary trees, Adv. Math. 223 (2010), no. 4, 1316-1355.

Cf. A000108, A201058, A268082.

Table[Denominator[Binomial[2n,n]/(2n)],{n,50}] (* Harvey P. Dale, Oct 04 2021 *)

a(n) = denominator(binomial(2*n,n)/(2*n)); \\ Michel Marcus, Jan 08 2024

a(n) = my(f = factor(2*n), res = 1); for(i = 1, #f~, v = val(2*n, f[i,1]) - 2*val(n, f[i, 1]) - f[i, 2]; if(v < 0, res*=f[i, 1]^(-v))); res
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Jan 10 2024

More terms from Michel Marcus, Jan 08 2024

A268083 Numbers k that are not prime powers and such that gcd(binomial(2*k-1,k), k) = 1.

Original entry on oeis.org

39, 55, 93, 111, 119, 155, 161, 185, 253, 275, 279, 305, 327, 333, 351, 363, 377, 403, 407, 413, 497, 511, 517, 533, 559, 629, 635, 649, 655, 685, 689, 697, 707, 741, 749, 755, 779, 785, 791, 813, 817, 849, 871, 893, 901, 905, 923, 981, 1003, 1011, 1027, 1043

Offset: 1

Michel Marcus, Jan 26 2016

nonn

It seems there is a typo in the Gua and Zeng link, it gives 175 instead of 185 as a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Victor J.W. Guo and Jiang Zeng, Factors of binomial sums from the Catalan triangle, Journal of Number Theory 130 (2010) 172-186.

Cf. A000961, A088218, A268082.

[n : n in [2..2000] | not IsPrimePower(n) and Gcd(Binomial(2*n-1,n), n) eq 1]; // Vincenzo Librandi, Jan 26 2016

Select[Range[2,1100],!PrimePowerQ[#]&&GCD[Binomial[2#-1,#],#]==1&] (* Harvey P. Dale, May 26 2020 *)

isok(n) = (n != 1) && !isprimepower(n) && (gcd(binomial(2*n-1,n), n) == 1);

from math import gcd
from sympy import factorint
A268083_list, b = [], 1
for n in range(1,10**4):
    if len(factorint(n)) > 1 and gcd(b,n) == 1:
        A268083_list.append(n)
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Jan 26 2016

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A201059 Denominator of binomial(2n,n)/(2n).

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A268083 Numbers k that are not prime powers and such that gcd(binomial(2*k-1,k), k) = 1.

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Mathematica

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