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A211419 Integral factorial ratio sequence: a(n) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!). 12
1, 10, 198, 4420, 104006, 2521260, 62300700, 1560167752, 39457579590, 1005490725148, 25776935824948, 664048851069240, 17175945353271068, 445775181599116600, 11602978540817349240, 302767701121286251920, 7917664916276259668550 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..16.

Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], J. London Math. Soc., Vol. 79, Issue 2 (2009),422-444.

F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functionsarXiv:math/0701362 [math.NT], 2007.

FORMULA

The o.g.f. sum {n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas).

From Peter Bala, Sep 29 2015: (Start)

a(n) = Sum_{i = 0..n} binomial(6*n,i)*binomial(5*n-i-1,n-i).

a(n) = [x^n] ( (1 + x)^6/(1 - x)^4 )^n.

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)

a(n) ~ 27^n/sqrt(2*Pi*n). - Ilya Gutkovskiy, Jul 31 2016

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

MAPLE

A211419 := n-> (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!):

seq(A211419(n), n=0..20);

# Using the ogf from Karol A. Penson and Jean-Marie Maillard:

u := 27*x-1: c := (u^3*((3*x*u)^(1/2)*(12+81*x)-u^2+216*x-7))^(1/3):

gf := ((c^2-2*c*u+27*u*(7-81*x)*x-4*u)/(6*c*u^2))^(1/2):

ser := series(gf, x, 8); # Peter Luschny, May 03 2018

MATHEMATICA

Table[(6 n)!*(2 n)!/((4 n)!*(3 n)!*n!), {n, 0, 16}] (* Michael De Vlieger, Oct 04 2015 *)

From Karol A. Penson and Jean-Marie Maillard, May 02 2018: (Start)

  The explicit form of the ogf is: 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))]. (End)

PROG

(PARI) a(n) = (6*n)!*(2*n)!/((4*n)!*(3*n)!*n!);

vector(30, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

(MAGMA) [Factorial(6*n) * Factorial(2*n) / (Factorial(4*n) * Factorial(3*n) * Factorial(n)): n in [0..20]]; // Vincenzo Librandi, May 03 2018

CROSSREFS

Cf. A061162, A061163, A182400, A211420, A211421.

Cf. A000984 (k = 0), A091527 (k = 1), A001448 (k = 2), A262732 (k = 3), A262733 (k = 5), A211421 (k = 6), A262738.

Sequence in context: A222499 A097127 A249846 * A001085 A079436 A285021

Adjacent sequences:  A211416 A211417 A211418 * A211420 A211421 A211422

KEYWORD

nonn,easy,changed

AUTHOR

Peter Bala, Apr 10 2012

STATUS

approved

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Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)