cp's OEIS Frontend

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

%I A211421 #55 Aug 09 2025 08:27:42
%S A211421 1,14,390,12236,404550,13777764,478273692,16825310040,597752648262,
%T A211421 21397472070260,770557136489140,27884297395587240,1013127645555452700,
%U A211421 36935287875280348776,1350441573221798941560,49498889739033621986736,1818284097150186829038150
%N A211421 Integral factorial ratio sequence: a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!).
%C A211421 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 A211421 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 A211421 Vincenzo Librandi, <a href="/A211421/b211421.txt">Table of n, a(n) for n = 0..600</a>
%H A211421 Peter Bala, <a href="/A100100/a100100_1.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%H A211421 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 A211421 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 A211421 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 A211421 From _Peter Bala_, Sep 29 2015: (Start)
%F A211421 a(n) = Sum_{i = 0..n} binomial(8*n,i) * binomial(7*n-i-1,n-i).
%F A211421 a(n) = [x^n] ( (1 + x)^8/(1 - x)^6 )^n.
%F A211421 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 A211421 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 A211421 a(n) ~ 2^(10*n)*27^(-n)/sqrt(2*Pi*n). - _Ilya Gutkovskiy_, Jul 31 2016
%F A211421 a(n) = (2^n/n!)*Product_{k = 3*n..4*n-1} (2*k + 1). - _Peter Bala_, Feb 26 2023
%F A211421 From _Seiichi Manyama_, Aug 09 2025: (Start)
%F A211421 a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(6*n)).
%F A211421 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(8*n,k) * binomial(2*n-k,n-k).
%F A211421 a(n) = Sum_{k=0..n} 2^k * binomial(6*n+k-1,k) * binomial(2*n-k,n-k).
%F A211421 a(n) = 4^n * binomial((8*n-1)/2,n).
%F A211421 a(n) = [x^n] 1/(1-4*x)^((6*n+1)/2).
%F A211421 a(n) = [x^n] (1+4*x)^((8*n-1)/2). (End)
%p A211421 #A211421
%p A211421 a := n -> (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);
%p A211421 seq(a(n), n = 0..16);
%t A211421 Table[(8 n)!*(3 n)!/((6 n)!*(4 n)!*n!), {n, 0, 15}] (* _Michael De Vlieger_, Oct 04 2015 *)
%o A211421 (PARI) a(n) = (8*n)!*(3*n)!/((6*n)!*(4*n)!*n!);
%o A211421 vector(30, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015
%o A211421 (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 A211421 Cf. A061162, A061163, A211419, A211420.
%Y A211421 Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A262740.
%K A211421 nonn,easy
%O A211421 0,2
%A A211421 _Peter Bala_, Apr 10 2012