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 A364175 #10 Jul 16 2023 05:52:03
%S A364175 1,36,3564,408408,49697388,6249195036,802241960520,104466877291260,
%T A364175 13746018177013356,1823169705017624880,243331037661693468564,
%U A364175 32641262295291161362656,4396944340992842923469640,594371374049863341847620936,80586283761263090599592845140
%N A364175 a(n) = (6*n)!*(2*n/3)!/((3*n)!*(2*n)!*(5*n/3)!).
%C A364175 A295445, defined by A295445(n) = (18*n)!*(2*n)! / ((9*n)!*(6*n)!*(5*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 15). Here we are essentially considering the sequence {A295445(n/3) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (2*n/3)! := Gamma(1 + 2*n/3).
%C A364175 This sequence is only conjecturally an integer sequence.
%C A364175 Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
%H A364175 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
%F A364175 a(n) ~ c^n * 1/sqrt(5*Pi*n) where c = (1296/25)*20^(1/3) = 140.7154092442799....
%F A364175 a(n) = 93312*(2*n - 3)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11)*(6*n - 13)*(6*n - 17)/(5*n*(n - 1)*(n - 2)*(5*n - 3)*(5*n - 6)*(5*n - 9)*(5*n - 12))*a(n-3) with a(0) = 1, a(1) = 36 and a(2) = 3564.
%p A364175 seq( simplify((6*n)!*(2*n/3)!/((3*n)!*(2*n)!*(5*n/3)!)), n = 0..15);
%Y A364175 Cf. A276100, A276101, A276102, A295431, A295445, A347854, A347855, A347856, A347857, A347858, A364172 - A364185.
%K A364175 nonn,easy
%O A364175 0,2
%A A364175 _Peter Bala_, Jul 13 2023