%I #35 Aug 10 2023 20:43:20

%S 1,40,4620,622336,89237148,13236695040,2005604901300,308350245273600,

%T 47913489552349980,7505566011722039296,1183237138556438547120,

%U 187495217080545878999040,29836408028165719837829700,4764790302634058161217077248

%N a(n) = (6*n)!/((3*n)!*(2*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)!*n!/((6*n)!*(4*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 1). See A295431. It is known that u(n) is integral and satisfies the congruences u(n*p) == u(n) ( mod p^3 ) for prime p >= 5 and any positive integer n (Zudilin, Section 5); the o.g.f. Sum_{n >= 0} u(n)*x^n is algebraic over Q(x) (Rodriguez-Villegas).

%C Here we are essentially considering the sequence ( u(n/2) )n>=0. The sequence is conjectured to be integral.

%H Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>

%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.

%H Wadim Zudilin, <a href="https://arxiv.org/abs/1901.07843">Congruences for q-binomial coefficients</a>, arXiv:1901.07843 [math.NT], 2019.

%F a(n) = binomial(6*n,2*n)*binomial(4*n,n)/binomial(3*n/2,n).

%F a(2*n) = A295431(n).

%F a(2*n) = 24*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)/( n*(2*n - 1)*(3*n - 1)*(3*n - 2) )*a(2*n-2);

%F a(2*n+1) = 96*(12*n + 1)*(12*n - 1)*(12*n + 5)*(12*n - 5)/( n*(2*n + 1)*(6*n + 1)*(6*n - 1) )*a(2*n-1).

%F Asymptotics: a(n) ~ 32^n/sqrt(6*Pi*n) * 3^(3*n/2) as n -> infinity.

%F O.g.f.: A(x) = hypergeom([1/12, 5/12, 7/12, 11/12], [1/3, 1/2, 2/3], 27648*x^2) + 40*x*hypergeom([11/12, 13/12, 7/12, 17/12], [3/2, 5/6, 7/6], 27648*x^2) is conjectured to be algebraic over Q(x).

%F Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k.

%F From _Karol A. Penson_, Feb 17 2023: (Start)

%F An integral representation of a(n) as the n-th power moment of the weight function W(x) is given by a(n) = Integral_{x=0..96*sqrt(3)} x^n*W(x), where W(x) = W_1(x) + W_2(x) + W_3(x) + W_4(x) and the functions W_n(x) are:

%F W_1(x) = sqrt(2)*3^(3/4)*hypergeom([1/12, 5/12, 7/12, 3/4], [1/6, 1/2, 2/3], x^2/27648)*Gamma(3/4)/(18*sqrt(Pi)*x^(5/6)*Gamma(2/3)*Gamma(7/12)).

%F W_2(x) = sqrt(2)*cos((5*Pi)/12)*Gamma(2/3)*csc(Pi/12)*Gamma(3/4)*3^(1/4)* hypergeom([5/12, 3/4, 11/12, 13/12], [1/2, 5/6, 4/3], x^2/27648)/(2304*Pi^(3/2)* Gamma(11/12)*x^(1/6)).

%F W_3(x) = cos((5*Pi)/12)*3^(1/4)*Gamma(11/12)*x^(1/6)*hypergeom([7/12, 11/12, 13/12, 5/4], [2/3, 7/6, 3/2], x^2/27648)/(3456*sqrt(Pi)*Gamma(2/3)*Gamma(3/4)).

%F W_4(x) = 7*sin((5*Pi)/12)*Gamma(2/3)*Gamma(7/12)*3^(3/4)*x^(5/6)*hypergeom([11/12, 5/4,17/12, 19/12], [4/3, 3/2, 11/6], x^2/27648))/(1327104*Pi^(3/2)*Gamma(3/4)).

%F The function W(x) is positive on the support x = (0..96*sqrt(3)) and is singular at both endpoints of the support. The function W(x) is unique as it is the solution of the Hausdorff moment problem. (End)

%e a(11) - a(1) = 187495217080545878999040 - 40 = (2^3)*(5^3)*(11^3)*140867931690868429 == 0 (mod 11^3).

%p seq(binomial(6*n,2*n)*binomial(4*n,n)/binomial(3*n/2,n), n = 0..13);

%o (Python)

%o from math import factorial

%o from sympy import factorial2

%o def A347854(n): return int((factorial(6*n)*factorial2(n)<<n)//(factorial(3*n)*factorial(n<<1)*factorial2(3*n))) # _Chai Wah Wu_, Aug 10 2023

%Y Cf. A295431, A347855, A347856, A347857, A347858.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Sep 16 2021