OFFSET
0,2
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
J. 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.
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math.NT/0701362, 2007.
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
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 *)
PROG
(Magma) [Factorial(20*n)*Factorial(n)/(Factorial(10*n)*Factorial(7*n)*Factorial(4*n)): n in [0..8]]; // Vincenzo Librandi, Oct 26 2011
(PARI) a(n)=(20*n)!*n!/(10*n)!/(7*n)!/(4*n)! \\ Charles R Greathouse IV, Apr 10 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard Stanley, Apr 17 2001
STATUS
approved