A182400 - OEIS

%I #48 Jul 21 2023 17:21:37

%S 1,28,1716,118864,8684340,653817528,50181947376,3903669874104,

%T 306689672988468,24278779897856848,1933612147959994216,

%U 154751222973374578656,12435284300689518633456,1002664938117354309314220,81080672610600385236492840,6573062133232532447808798864

%N Integral factorial ratio sequence: a(n) = (2*n)!*(8*n)!/(n!*(4*n)!*(5*n)!).

%C For any nonnegative integers m, n the ratio (2*m)!*(2*n)!/(m!*(m+n)!*n!) provides an integer (theorem attributed to Catalan, see Umberto Scarpis in References), and this sequence is the case m = 4*n.

%D Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.

%H Bruno Berselli, <a href="/A182400/b182400.txt">Table of n, a(n) for n = 0..100</a>

%H Jonathan W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, 2007, arXiv:0709.1977v1 [math.NT], page 3 (Theorem 1.2, formula 8: case a=1, b=4).

%H Alexander Borisov, <a href="https://arxiv.org/abs/math/0505167">Quotient singularities, integer ratios of factorials and the Riemann Hypothesis</a>, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).

%F a(n) = (-1024)^n*binomial(4*n-1/2,5*n).

%F From _Ilya Gutkovskiy_, Jan 31 2017: (Start)

%F G.f.: 5F4(1/8,3/8,1/2,5/8,7/8; 1/5,2/5,3/5,4/5; 262144*x/3125).

%F E.g.f.: 5F5(1/8,3/8,1/2,5/8,7/8; 1/5,2/5,3/5,4/5,1; 262144*x/3125).

%F a(n) ~ 2^(18*n+1/2)/(sqrt(Pi*n)*5^(5*n+1/2)). (End)

%F a(n) = a(n-1)*32*(2*n - 1)*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/(5*n*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)). - _Neven Sajko_, Jul 21 2023

%t Table[((2 n)! (8 n)!)/(n! (4 n)! (5 n)!), {n, 0, 15}]

%o (Magma)

%o [Factorial(2*m)*Factorial(2*n)/(Factorial(m)*Factorial(m+n)*Factorial(n)) where m is 4*n: n in [0..15]];

%o (Maxima) makelist((-1024)^n*binomial(4*n-1/2,5*n),n,0,15);

%Y Cf. A000984 (m = n), A005810 (m = 2*n), A211419 (m = 3*n).

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Apr 27 2012