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 A364184 #19 Aug 28 2025 07:30:53
%S A364184 1,1408,6374082,32993443840,180669266788650,1020694137466257408,
%T A364184 5882199787281395215344,34369110490167819009785856,
%U A364184 202857467914154836183288657770,1206640354461153104738279049134080,7221430962039777689508936047385667332
%N A364184 a(n) = (12*n)!*(2*n)!*(3*n/2)!/((6*n)!*(9*n/2)!*(4*n)!*n!).
%C A364184 A295481, defined by A295481(n) = (24*n)!*(4*n)!*(3*n)! / ((12*n)!*(9*n)!*(8*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 51). Here we are essentially considering the sequence {A295481(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! := Gamma(1 + 9*n/2).
%C A364184 This sequence is only conjecturally an integer sequence.
%C A364184 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 A364184 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 A364184 a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (2^15)/(3^2) * sqrt(3).
%F A364184 a(n) = 49152*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(n*(n - 1)*(9*n - 2)*(9*n - 4)*(9*n - 8)*(9*n - 10)*(9*n - 14)*(9*n - 16))*a(n-2) with a(0) = 1 and a(1) = 1408.
%p A364184 seq( simplify((12*n)!*(2*n)!*(3*n/2)!/((6*n)!*(9*n/2)!*(4*n)!*n!)), n = 0..15);
%Y A364184 Cf. A276100, A276101, A276102, A295431, A295481, A347854, A347855, A347856, A347857, A347858, A364173 - A364183.
%K A364184 nonn,easy,changed
%O A364184 0,2
%A A364184 _Peter Bala_, Jul 13 2023