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%I #12 Jul 16 2023 05:49:26
%S 1,384,461890,638582784,935387159850,1414457284624384,
%T 2182519096151533552,3414991108739243704320,5398397695681095146608490,
%U 8600772808890306913527398400,13787702861800799166026014363140,22213518902232966637201617101783040,35936545440404705429404600374145350960
%N a(n) = (10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!).
%C A295475, defined by A295475(n) = (20*n)!*(3*n)! / ((10*n)!*(9*n)!*(4*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 45). Here we are essentially considering the sequence {A295475(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*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 = (2^11)*(5^5)/(3^8)*sqrt(3).
%F a(n) = 409600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(27*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) = 384
%p seq( simplify((10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!)), n = 0..15);
%Y Cf. A276100, A276101, A276102, A295431, A295475, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jul 13 2023
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