A182400 Integral factorial ratio sequence: a(n) = (2*n)!*(8*n)!/(n!*(4*n)!*(5*n)!).

1, 28, 1716, 118864, 8684340, 653817528, 50181947376, 3903669874104, 306689672988468, 24278779897856848, 1933612147959994216, 154751222973374578656, 12435284300689518633456, 1002664938117354309314220, 81080672610600385236492840, 6573062133232532447808798864

Offset: 0

Author

Bruno Berselli, Apr 27 2012

Keywords

• nonn
• easy

Comments

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.

References

• 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.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..100
• Jonathan W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], page 3 (Theorem 1.2, formula 8: case a=1, b=4).
• Alexander Borisov, Quotient singularities, integer ratios of factorials and the Riemann Hypothesis, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).

Crossrefs

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

Programs

Magma

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

Mathematica

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

Maxima

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

Formula

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

From Ilya Gutkovskiy, Jan 31 2017: (Start)

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

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).

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

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