A061164 a(n) = (20*n)!n!/((10*n)!(7*n)!(4*n)!).

1, 5542680, 190818980609400, 7691041400616850556280, 330014847932376708502470210680, 14647137653300940580784413641872332680, 663999280578266939183818080578580843597787800, 30541460340748361003270983719744457382865889296237000

Offset: 0

Author

Richard Stanley, Apr 17 2001

Keywords

• easy
• nonn

Comments

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.

References

• M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..22
• Jonathan W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.
• M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22 p. 11.
• Fernando Rodriguez Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Crossrefs

Cf. A061162, A061163.

Programs

Magma

[Factorial(20*n)*Factorial(n)/(Factorial(10*n)*Factorial(7*n)*Factorial(4*n)): n in [0..8]]; // Vincenzo Librandi, Oct 26 2011

Maple

A061164 := proc(n)
        binomial(20*n,10*n)*binomial(10*n,3*n)/binomial(4*n,n) ;
end proc:
seq(A061164(n),n=0..10) ; # R. J. Mathar, Oct 26 2011

Mathematica

Table[((20n)!n!)/((10n)!(7n)!(4n)!),{n,0,10}] (* Harvey P. Dale, Oct 25 2011 *)

PARI

a(n)=(20*n)!*n!/(10*n)!/(7*n)!/(4*n)! \\ Charles R Greathouse IV, Apr 10 2012

Formula

One of the 52 sporadic integral factorial ratio sequences found by V. I. Vasyunin (see Bober, Table 2, Entry 43). The o.g.f. sum {n >= 1} a(n)*z^n is an algebraic function over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012

O.g.f. is a generalized hypergeometric function 8F7([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/7, 2/7, 3/7, 1/2, 4/7, 5/7, 6/7], ((2^22)*(5^10)*x)/7^7). - Karol A. Penson, Apr 11 2022

a(n) ~ 2^(22*n - 1) * 5^(10*n) / (sqrt(Pi*n) * 7^(7*n + 1/2)). - Vaclav Kotesovec, Aug 27 2024