A364176 - OEIS

%I #10 Jul 16 2023 05:52:22

%S 1,7168,168043980,4488240824320,126694219977836700,

%T 3688258943632086663168,109504706026534324525391988,

%U 3295939064766794222800490987520,100204869963549181630558779565943580,3070025447039504554088467623457608171520,94632263448378916462441320194245442445186480

%N a(n) = (15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(6*n)!*(5*n)!*n!).

%C A295456, defined by A295456(n) = (30*n)!*(5*n)!*(4*n)! / ((15*n)!*(12*n)!*(10*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 26). Here we are essentially considering the sequence {A295456(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 This sequence is only conjecturally an integer sequence.

%C 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 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 a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = 18750*sqrt(3).

%F a(n) = 4800*(15*n - 1)*(15*n - 7)*(15*n - 11)*(15*n - 13)*(15*n - 17)*(15*n - 19)*(15*n - 23)*(15*n - 29)/(n*(n - 1)*(3*n - 2)*(3*n - 4)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11))*a(n-2) with a(0) = 1 and a(1) = 7168.

%p seq( simplify((15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(6*n)!*(5*n)!*n!)), n = 0..15)

%Y Cf. A276100, A276101, A276102, A295431, A295456, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 13 2023