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%I #20 Jul 30 2022 05:50:06
%S 1,9,189,4620,120285,3241134,89237148,2493521172,70429218525,
%T 2005604901300,57481750139814,1656023714623980,47913489552349980,
%U 1391243084942932620,40519970408738302020,1183237138556438547120
%N a(n) = (4*n)!/((2*n)!*(n)!) * (n/3)!/(4*n/3)!.
%C Fractional factorials are defined using the Gamma function; for example, (n/3)! := Gamma(1 + n/3). 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. Here we are essentially considering the sequence (u(n/3))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(4*n,2*n)*binomial(2*n,n)/binomial(4*n/3,n).
%F a(3*n) = A295431.
%F D-finite with recurrence -n*(n-1)*(n-2)*(2*n-3)*a(n) + 216*(4*n-11)*(4*n-1)*(4*n-5)*(4*n-7)*a(n-3).
%F Asymptotics: a(n) ~ 1/(2*sqrt(Pi*n))*2^(10*n/3)*3^n as n -> infinity.
%F O.g.f.: A(x) = hypergeom([11/12, 7/12, 5/12, 1/12], [2/3, 1/2, 1/3], 27648*x^3) + 9*x*hypergeom([11/12, 5/4, 5/12, 3/4], [5/6, 4/3, 2/3], 27648*x^3) + 189*x^2*hypergeom([19/12, 13/12, 5/4, 3/4], [7/6, 5/3, 4/3], 27648*x^3) 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) = 1656023714623980 - 9 = (3^2)*7*(11^3)*17* 1161713471 == 0 (mod 11^3).
%p seq( (4*n)!/((2*n)!*(n)!) * GAMMA(1+n/3)/GAMMA(1+4*n/3), n = 0..15);
%t Table[Binomial[4n,2n] Binomial[2n,n]/Binomial[4 n/3,n],{n,0,20}] (* _Harvey P. Dale_, Apr 09 2022 *)
%Y Cf. A259431, A347854, A347855, A347856, A347857, A347858.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Sep 17 2021
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