A263931 - OEIS

%I #27 Sep 07 2022 12:31:58

%S 1,1,2,4,2,36,12,24,90,20,4,168,28,1400,5400,720,90,5940,23100,46200,

%T 180180,17160,1560,140400,11700,45864,179928,13328,52360,5969040,

%U 397936,795872,3133746,12345060,726180,2863224,159068,318136,1255800,4958800,247940

%N a(n) = binomial(2*n, n) / Product(p prime | n < p <= 2*n).

%C The highest exponent in the prime factorization of a(n) is A263922(n), n>=2.

%C a(n) is even for n>=2.

%C By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree. - _Robert FERREOL_, Sep 06 2022

%H David A. Corneth, <a href="/A263931/b263931.txt">Table of n, a(n) for n = 0..5806</a> (terms < 10^1000)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html">Erdős Squarefree Conjecture</a>.

%F a(n) = A000984(n)/A261130(n).

%p a := n -> binomial(2*n,n)/convert(select(isprime, {$n+1..2*n}),`*`):

%p seq(a(n), n=0..40);

%o (PARI) a(n) = { my(res = 1); forprime(p = 2, n, res*= p^(val(2*n, p) - 2*val(n, p))); forprime(p = n + 1, 2*n, res*= p^(val(2*n, p) - 2*val(n, p) - 1)); res } val(n, p) = my(r=0); while(n, r+=n\=p);r \\ _David A. Corneth_, Apr 03 2021

%o (Python)

%o from math import comb

%o from sympy import primorial

%o def A263931(n): return comb(m:=n<<1,n)*primorial(n,nth=False)//primorial(m,nth=False) if n else 1 # _Chai Wah Wu_, Sep 07 2022

%Y Cf. A000984, A261130, A263922, A356637.

%K nonn

%O 0,3

%A _Peter Luschny_, Oct 31 2015