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A364182 a(n) = (12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!).

%I A364182 #17 Jul 18 2023 04:08:46
%S A364182 1,7392,267711444,11489451294720,527048385075849780,
%T A364182 25051434899696246587392,1217325447549161369383451760,
%U A364182 60050961586064738516089033457664,2994861478939539397101967737771147060,150602318360773064327512837557840362078208
%N A364182 a(n) = (12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!).
%C A364182 A295477, defined by A295477(n) = (24*n)!*n! / ((12*n)!*(8*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 47). Here we are essentially considering the sequence {A295477(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (5*n/2)! := Gamma(1 + 5*n/2).
%C A364182 This sequence is only conjecturally an integer sequence.
%C A364182 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 A364182 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 A364182 a(n) ~ c^n * 1/sqrt(20*Pi*n), where c = (2^12)*(3^6)/(5^3) * sqrt(5).
%F A364182 a(n) = 82944*(12*n - 1)*(12*n - 5)(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(5*n*(n - 1)*(2*n - 1)*(2*n - 3)*(5*n - 2)*(5*n - 4)*(5*n - 6)*(5*n - 8))*a(n-2) with a(0) = 1 and a(1) = 7392
%p A364182 seq( simplify((12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!)), n = 0..15);
%Y A364182 Cf. A276100, A276101, A276102, A295431, A295477, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.
%K A364182 nonn,easy
%O A364182 0,2
%A A364182 _Peter Bala_, Jul 13 2023