This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263931 #46 Dec 20 2024 09:24:46
%S A263931 1,1,2,4,2,36,12,24,90,20,4,168,28,1400,5400,720,90,5940,23100,46200,
%T A263931 180180,17160,1560,140400,11700,45864,179928,13328,52360,5969040,
%U A263931 397936,795872,3133746,12345060,726180,2863224,159068,318136,1255800,4958800,247940
%N A263931 a(n) = binomial(2*n, n) / Product(p prime | n < p <= 2*n).
%C A263931 The highest exponent in the prime factorization of a(n) is A263922(n), n>=2.
%C A263931 a(n) is even for n>=2.
%C A263931 By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree. - _Robert FERREOL_, Sep 06 2022
%C A263931 Conjecture: n=5 is the only n for which a(n)^2 > binomial(2*n,n). - _Richard Peterson_, Dec 18 2024
%H A263931 David A. Corneth, <a href="/A263931/b263931.txt">Table of n, a(n) for n = 0..5806</a> (terms < 10^1000)
%H A263931 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html">Erdős Squarefree Conjecture</a>.
%F A263931 a(n) = A000984(n)/A261130(n).
%p A263931 a := n -> binomial(2*n,n)/convert(select(isprime, {$n+1..2*n}),`*`):
%p A263931 seq(a(n), n=0..40);
%o A263931 (PARI)
%o A263931 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 }
%o A263931 val(n, p) = my(r=0); while(n, r+=n\=p);r \\ _David A. Corneth_, Apr 03 2021
%o A263931 (Python)
%o A263931 from math import comb
%o A263931 from sympy import primorial
%o A263931 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 A263931 Cf. A000984, A261130, A263922, A356637.
%K A263931 nonn
%O A263931 0,3
%A A263931 _Peter Luschny_, Oct 31 2015