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A211419 a(n) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!).

%I A211419 #122 Aug 09 2025 07:50:22
%S A211419 1,10,198,4420,104006,2521260,62300700,1560167752,39457579590,
%T A211419 1005490725148,25776935824948,664048851069240,17175945353271068,
%U A211419 445775181599116600,11602978540817349240,302767701121286251920,7917664916276259668550,207452338901630123085180
%N A211419 a(n) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!).
%C A211419 This sequence is the particular case a = 3, b = 2 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), A211420 (a = 4, b = 1), A211421 (a = 4, b = 3) and A061163 (a = 5, b = 1).
%C A211419 This is the case m = 3n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - _Bruno Berselli_, Apr 27 2012
%C A211419 Sequence terms are given by the coefficient of x^n in the expansion of ((1 + x)^(k+2)/(1 - x)^k)^n when k = 4. See the cross references for related sequences obtained from other values of k. - _Peter Bala_, Sep 29 2015
%D A211419 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.
%H A211419 Seiichi Manyama, <a href="/A211419/b211419.txt">Table of n, a(n) for n = 0..500</a>
%H A211419 Peter Bala, <a href="/A100100/a100100_1.pdf">Notes on logarithmic differentiation, the binomial transform and series reversion</a>
%H A211419 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977v1 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2 (2009), 422-444.
%H A211419 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 A211419 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 A211419 From _Peter Bala_, Sep 29 2015: (Start)
%F A211419 a(n) = Sum_{i = 0..n} binomial(6*n, i) * binomial(5*n-i-1, n-i).
%F A211419 a(n) = [x^n] ( (1 + x)^6/(1 - x)^4 )^n.
%F A211419 O.g.f. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 10*x + 149*x^2 + 2630*x^3 + 51002*x^4 + ... has integer coefficients and equals 1/x * series reversion of x*(1 - x)^4/ (1 + x)^6. See A262738. (End)
%F A211419 a(n) ~ 27^n/sqrt(2*Pi*n). - _Ilya Gutkovskiy_, Jul 31 2016
%F A211419 O.g.f.: sqrt((4 + 7290*x^2 - 59049*x^3 + 2*(8 + 3*sqrt(3)*sqrt(x*(-1 + 27*x)^7*(4 + 27*x)^2) - 27*x*(-2 + 27*x)*(-17 + 27*x*(19 + 27*x*(-11 + 27*x))))^(1/3) + (8 + 3*sqrt(3)*sqrt(x*(-1 + 27*x)^7*(4 + 27*x)^2) - 27*x*(-2 + 27*x)*(-17 + 27*x*(19 + 27*x*(-11 + 27*x))))^(2/3) - 27*x*(11 + 2*(8 + 3*sqrt(3)*sqrt(x*(-1 + 27*x)^7*(4 + 27*x)^2) - 27*x*(-2 + 27*x)*(-17 + 27*x*(19 + 27*x*(-11 + 27*x))))^(1/3)))/(6*(1 - 27*x)^2*(8 + 3*sqrt(3)*sqrt(x*(-1 + 27*x)^7*(4 + 27*x)^2) - 27*x*(-2 + 27*x)*(-17 + 27*x*(19 + 27*x*(-11 + 27*x))))^(1/3))). - _Karol A. Penson_ and Jean-Marie Maillard, May 02 2018
%F A211419 Right-hand side of the binomial sum identity: Sum_{k = 0..2*n} (-1)^(n+k) * binomial(6*n, 2*n+k) * binomial(2*n, k) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!). - _Peter Bala_, Jan 19 2020
%F A211419 a(n) = 6*(6*n - 1)*(2*n - 1)*(6*n - 5)*a(n-1)/(n*(4*n - 1)*(4*n - 3)). - _Neven Sajko_, Jul 19 2023
%F A211419 From _Peter Luschny_, Feb 22 2024: (Start)
%F A211419 a(n) = 4^n*(Gamma(3*n + 1/2)/Gamma(2*n + 1/2))/Gamma(n + 1).
%F A211419 O.g.f.: hypergeom([1/6, 1/2, 5/6], [1/4, 3/4], 27*z). (End)
%F A211419 From _Seiichi Manyama_, Aug 09 2025: (Start)
%F A211419 a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(4*n)).
%F A211419 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(6*n,k) * binomial(2*n-k,n-k).
%F A211419 a(n) = Sum_{k=0..n} 2^k * binomial(4*n+k-1,k) * binomial(2*n-k,n-k).
%F A211419 a(n) = 4^n * binomial((6*n-1)/2,n).
%F A211419 a(n) = [x^n] 1/(1-4*x)^((4*n+1)/2).
%F A211419 a(n) = [x^n] (1+4*x)^((6*n-1)/2). (End)
%p A211419 A211419 := n-> (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!):
%p A211419 seq(A211419(n), n=0..20);
%p A211419 # Using the o.g.f. from _Karol A. Penson_ and Jean-Marie Maillard:
%p A211419 u := 27*x-1: c := (u^3*((3*x*u)^(1/2)*(12+81*x)-u^2+216*x-7))^(1/3):
%p A211419 gf := ((c^2-2*c*u+27*u*(7-81*x)*x-4*u)/(6*c*u^2))^(1/2):
%p A211419 ser := series(gf, x, 8); # _Peter Luschny_, May 03 2018
%p A211419 ogf := hypergeom([1/6, 1/2, 5/6], [1/4, 3/4], 27*z): ser := series(ogf, z, 20):
%p A211419 seq(coeff(ser, z, n), n = 0..17);  # _Peter Luschny_, Feb 22 2024
%t A211419 Table[(6 n)!*(2 n)!/((4 n)!*(3 n)!*n!), {n, 0, 16}] (* _Michael De Vlieger_, Oct 04 2015 *)
%t A211419 CoefficientList[Series[Sqrt[(4 + 7290 x^2 - 59049 x^3 + 2 (8 + 3 Sqrt[3] Sqrt[x (-1 + 27 x)^7 (4 + 27 x)^2] - 27 x (-2 + 27 x) (-17 + 27 x (19 + 27 x (-11 + 27 x))))^(1/3) + (8 + 3 Sqrt[3] Sqrt[x (-1 + 27 x)^7 (4 + 27 x)^2] - 27 x (-2 + 27 x) (-17 + 27 x (19 + 27 x (-11 + 27 x))))^(2/3) - 27 x (11 + 2 (8 + 3 Sqrt[3] Sqrt[x (-1 + 27 x)^7 (4 + 27 x)^2] - 27 x (-2 + 27 x) (-17 + 27 x (19 + 27 x (-11 + 27 x))))^(1/3)))/(6 (1 - 27 x)^2 (8 + 3 Sqrt[3] Sqrt[x (-1 + 27 x)^7 (4 + 27 x)^2] - 27 x (-2 + 27 x) (-17 + 27 x (19 + 27 x (-11 + 27 x))))^(1/3))],{x,0,16}],x] (* _Karol A. Penson_ and Jean-Marie Maillard, May 02 2018 *)
%o A211419 (PARI) a(n) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!);
%o A211419 vector(30, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015
%o A211419 (Magma) [Factorial(6*n) * Factorial(2*n) / (Factorial(4*n) * Factorial(3*n) * Factorial(n)): n in [0..20]]; // _Vincenzo Librandi_, May 03 2018
%Y A211419 Cf. A061162, A061163, A182400, A211420, A211421.
%Y A211419 Cf. A000984 (k=0), A091527 (k=1), A001448 (k=2), A262732 (k=3), A262733 (k=5), A211421 (k=6), A262738.
%K A211419 nonn,easy
%O A211419 0,2
%A A211419 _Peter Bala_, Apr 10 2012