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 A370601 #28 Jan 08 2025 17:02:31
%S A370601 0,1,3,23,37,156,371,1207,2826,8738,14839,60738,80177,148702,466545,
%T A370601 1673039,2077633,4771287,10665251,26790730,72170979,212182718,
%U A370601 248771227,1074691776,1488526850,2533202074,8444006973,13950326222,18313391221,74263958970,164820521219
%N A370601 a(n) is the number of integer values of the function F_2n(x) = Product_{i=0..2n-1} (x + i) / Sum_{i=0..2n-1} (x + i).
%C A370601 If k is an odd prime then F_k(x) = Product_{i=0..k} (x + i)/ Sum_{i=0..k} (x + i) is not an integer if and only if x==(k+1)/2 (mod k). If k is odd but not prime, then F_k(x) is an integer for all positive integers x. On the other hand, for every even k, there is only a finite number of integer values of F_k(x).
%H A370601 Vladimir Letsko, <a href="https://youtu.be/vR4RTilBQno">Math. vertical</a>, YouTube video, 2023 (in Russian).
%F A370601 a(n) = tau(d/gcd(d, n)) - n, where d = ((2*n-1)!!)^2 and tau(k) is the number of divisors of the positive integer k.
%e A370601 a(3) = 3 because F_6(x) has exactly 3 integer values: F_6(5) = 3360, F_6(10) = 48048, and F_6(35) = 12282816.
%p A370601 a := proc(n) local d; d := doublefactorial(2*n-1)^2; numtheory[tau](d/igcd(n, d)) - n end: seq(a(n), n = 1..31);
%t A370601 Table[DivisorSigma[0, (2*n - 1)!!^2/GCD[(2*n - 1)!!^2, n]] - n, {n, 1, 30}] (* _Vaclav Kotesovec_, Feb 23 2024 *)
%Y A370601 Cf. A000005, A001818, A126695.
%K A370601 nonn
%O A370601 1,3
%A A370601 _Vladimir Letsko_, Feb 23 2024