A211420 - OEIS

%I #31 Aug 27 2024 04:13:47

%S 1,140,60060,29745716,15628090140,8480843582640,4697400936504900,

%T 2638798257262351800,1497753729733989900060,856840435680656569701776,

%U 493243073668546377605912560,285369375758780754651194529300,165789876049841088844342275759300

%N a(n) = (8*n)!*n!/((4*n)!*(3*n)!*(2*n)!).

%C This sequence is the particular case a = 4, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and GCD(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A061162 (a = 3, b = 1), A211419(a = 3, b = 2), A211421(a = 4, b = 3) and A061163 (a = 5, b = 1).

%D R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

%H J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, 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 The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas).

%F D-finite with recurrence: 3*(3*n-1)*(2*n-1)*(3*n-2)*n*a(n) - 8*(8*n-3)*(8*n-1)*(8*n-7)*(8*n-5)*a(n-1) = 0. - _Georg Fischer_, Nov 30 2022

%F From _Peter Bala_, Jul 10 2023: (Start)

%F a(n) = Sum_{k = 0..3*n} binomial(8*n, k)*binomial(5*n-k-1, 3*n-k).

%F a(n) = [x^(3*n)] F(x)^n, where F(x) = (1 + x)^8/(1 - x)^2.

%F It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/3*( 1/(1 - t*F(x^(1/3))) +  1/(1 - t*F(w*x^(1/3))) +  1/(1 - t*F(w^2*x^(1/3))) ), where w = exp(2*Pi*i/3), and hence A(x), as stated above, is algebraic over Q(x) by Stanley 1999, Theorem 6.33, p. 197. (End)

%F From _Karol A. Penson_, Feb 23 2024: (Start)

%F O.g.f.: hypergeometric4F3([1/8, 3/8, 5/8, 7/8], [1/3, 1/2, 2/3], (2^14*z)/27). (O.g.f.(z))^2 satisfies the algebraic equation of order 16, in which the powers of (o.g.f.(z))^2 are multiplied by polynomials p(n, z) with integer coefficients, in the form: Sum_{n = 0..16} p(n, z) * (o.g.f.(z))^(2*n) = 0.

%F Here is the list of orders, in the variable z, of all polynomials p(n, z) for n = 0..16: 8,8,8,8,9,9,9,9,10,10,10,11,11,11,11,11,12. For example p(14, z) = 6*(2^13*z + 31*27)*(2^14*z - 27)^10. (End)

%F a(n) ~ 2^(14*n - 1/2) / (3^(3*n + 1/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 27 2024

%p a := n -> (2^(6*n)*GAMMA(4*n + 1/2))/(GAMMA(n + 1/2)*GAMMA(3*n + 1)):

%p seq(a(n), n = 0..12); # _Peter Luschny_, Jul 11 2023

%t Table[ 2^(6*n) * Gamma[4*n + 1/2] / (Gamma[n + 1/2] * Gamma[3*n + 1]), {n, 0, 12}] (* _James C. McMahon_, Feb 24 2024 *)

%Y Cf. A061162, A061163, A211419, A211421.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Apr 10 2012