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%I #17 Aug 10 2023 17:54:10
%S 1,20,990,56576,3432198,215147520,13768454700,893768826880,
%T 58626071754822,3876225891958784,257898242928604740,
%U 17245961314149335040,1158088115444301759900,78040346182201091555328
%N a(n) = (6*n)!/((4*n)!*n!) * (n/2)!/(3*n/2)!.
%C Fractional factorials are defined using the Gamma function; for example, (n/2)! := Gamma(1 + n/2).
%C The sequence defined by u(n) = (12*n)!/((8*n)!*(2*n)!) * n!/(3*n)! is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 3). See A295433. Here we are essentially considering the sequence (u(n/2))n>=0. The sequence is conjectured to be integral.
%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., Vol. 79, Issue 2 (2009), 422-444.
%H F. Rodriguez-Villegas, <a href="http://arxiv.org/abs/math/0701362">Integral ratios of factorials and algebraic hypergeometric functions</a>, arXiv:math/0701362 [math.NT], 2007.
%F a(n) = binomial(6*n,4*n)*binomial(2*n,n)/binomial(3*n/2,n).
%F a(2*n) = A295433(n).
%F a(2*n) = 54*(6*n-1)*(6*n-5)*(12*n-1)*(12*n-5)*(12*n-7)*(12*n-11)/(n*(2*n-1)*(8*n-1)*(8*n-3)*(8*n-5)*(8*n-7))*a(2*n-2);
%F a(2*n+1) = 216*(9*n^2-1)*(144*n^2-1)*(144*n^2-5^2)/(n*(2*n+1)*(64*n^2-1)*(64*n^2-3^2))*a(2*n-1).
%F Asymptotics: a(n) ~ 1/(2*sqrt(n*Pi)) *3^(9*n/2)/2^n as n -> infinity.
%F O.g.f. A(x) = hypergeom([1/12, 1/6, 5/12, 7/12, 5/6, 11/12], [1/8, 3/8, 1/2, 5/8, 7/8], (19683/4)*x^2) + 20*x*hypergeom([17/12, 13/12, 11/12, 7/12, 4/3, 2/3], [11/8, 9/8, 7/8, 5/8, 3/2], (19683/4)*x^2) is conjectured to be algebraic over Q(x).
%F Conjectural congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k.
%e Congruence: a(11) - a(1) = 17245961314149335040 - 20 = (2^2)*5*(11^3)*103*6289876694707 == 0 (mod 11^3).
%p seq( (6*n)!/((4*n)!*n!) * GAMMA(1+n/2)/GAMMA(1+3*n/2), n = 0..12);
%o (Python)
%o from math import factorial
%o from sympy import factorial2
%o def A347856(n): return int((factorial(6*n)*factorial2(n)<<n)//(factorial(n<<2)*factorial(n)*factorial2(3*n))) # _Chai Wah Wu_, Aug 10 2023
%Y Cf. A295433, A347854 - A347858.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Sep 17 2021
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