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A211421 Integral factorial ratio sequence: a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!). 14

%I #48 Feb 28 2023 08:19:13

%S 1,14,390,12236,404550,13777764,478273692,16825310040,597752648262,

%T 21397472070260,770557136489140,27884297395587240,1013127645555452700,

%U 36935287875280348776,1350441573221798941560,49498889739033621986736,1818284097150186829038150

%N Integral factorial ratio sequence: a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!).

%C This sequence is the particular case a = 4, b = 3 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) and A211420 (a = 4, b = 1).

%C Sequence terms are given by the coefficient of x^n in the expansion of ( (1 + x)^(k+2)/(1 - x)^k )^n when k = 6. See the cross references for related sequences obtained from other values of k. - _Peter Bala_, Sep 29 2015

%H Vincenzo Librandi, <a href="/A211421/b211421.txt">Table of n, a(n) for n = 0..600</a>

%H P. Bala, <a href="/A100100/a100100_1.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>

%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:0709.1977v1 [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 From _Peter Bala_, Sep 29 2015: (Start)

%F a(n) = Sum_{i = 0..n} binomial(8*n,i)*binomial(7*n-i-1,n-i).

%F a(n) = [x^n] ( (1 + x)^8/(1 - x)^6 )^n.

%F a(0) = 1 and a(n) = 2*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/( n*(6*n - 1)*(6*n - 3)*(6*n - 5) ) * a(n-1) for n >= 1.

%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 14*x + 293*x^2 + 7266*x^3 + 197962*x^4 + 5726364*x^5 + ... has integer coefficients and equals 1/x * series reversion of x*(1 - x)^6/(1 + x)^8. See A262740. (End)

%F a(n) ~ 2^(10*n)*27^(-n)/sqrt(2*Pi*n). - _Ilya Gutkovskiy_, Jul 31 2016

%F a(n) = (2^n/n!)*Product_{k = 3*n..4*n-1} (2*k + 1). - _Peter Bala_, Feb 26 2023

%p #A211421

%p a := n -> (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);

%p seq(a(n), n = 0..16);

%t Table[(8 n)!*(3 n)!/((6 n)!*(4 n)!*n!), {n, 0, 15}] (* _Michael De Vlieger_, Oct 04 2015 *)

%o (PARI) a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);

%o vector(30, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015

%o (Magma) [Factorial(8*n)*Factorial(3*n)/(Factorial(6*n)*Factorial(4*n)*Factorial(n)): n in [0..20]]; // _Vincenzo Librandi_, Aug 01 2016

%Y Cf. A061162, A061163, A211419, A211420.

%Y Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A262740.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Apr 10 2012

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Last modified March 28 14:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)