A091527 a(n) = ((3*n)!/n!^2)*(Gamma(1+n/2)/Gamma(1+3n/2)).

1, 4, 30, 256, 2310, 21504, 204204, 1966080, 19122246, 187432960, 1848483780, 18320719872, 182327718300, 1820797698048, 18236779032600, 183120225632256, 1842826521244230, 18581317012684800, 187679234340049620, 1898554215471513600, 19232182592635611060

Offset: 0

Author

Michael Somos, Jan 18 2004

Keywords

• nonn
• easy

Comments

Sequence terms are given by [x^n] ( (1 + x)^(k+2)/(1 - x)^k )^n for k = 1. See the crossreferences for related sequences obtained from other values of k. - Peter Bala, Sep 29 2015

Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for n >= 0. This is the case a = 1, b = 0. - Peter Bala, Aug 28 2016

References

• R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..985
• Peter Bala, Some integer ratios of factorials
• Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3.
• Karl Dilcher, Armin Straub, and Christophe Vignat, Identities for Bernoulli polynomials related to multiple Tornheim zeta functions, arXiv:1903.11759 [math.NT], 2019. See p. 11.
• I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
• Bernhard Heim and Markus Neuhauser, Asymptotic Distribution of the Zeros of recursively defined Non-Orthogonal Polynomials, arXiv:2107.05013 [math.CA], 2021.
• W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.

Crossrefs

Cf. A061162(n) = a(2n), A007297, A000984 (k = 0), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A211421 (k = 6), A276098, A276099.

Programs

Maple

a := n -> 4^n * `if`(n<2, 1, (2*(n+1)*binomial((3*n-1)/2, n + 1))/(n-1)):
seq(a(n), n=0..18); # Peter Luschny, Feb 03 2020

Mathematica

Table[((3 n)!/n!^2) Gamma[1 + n/2]/Gamma[1 + 3 n/2], {n, 0, 18}] (* Michael De Vlieger, Oct 02 2015 *)
Table[4^n Sum[Binomial[k - 1 + (n - 1)/2, k], {k, 0, n}], {n, 0, 18}] (* Michael De Vlieger, Aug 28 2016 *)

Maxima

B(x):=(-1/3+(2/3)*sqrt(1+9*x)*sin((1/3)*asin((2+27*x+54*x^2)/2/(1+9*x)^(3/2))))/x-1;
taylor(x*diff(B(x),x)/B(x),x,0,10); /* Vladimir Kruchinin, Oct 02 2015 */

PARI

a(n)=4^n*sum(i=0,n,binomial(i-1+(n-1)/2,i))

PARI

vector(30, n, sum(k=0, n, binomial(3*n-3, k)*binomial(2*n-k-3, n-k-1))) \\ Altug Alkan, Oct 04 2015

Python

from math import factorial
from sympy import factorial2
def A091527(n): return int((factorial(3*n)*factorial2(n)<Chai Wah Wu, Aug 10 2023

Formula

D-finite with recurrence n*(n - 1)*a(n) = 12*(3*n - 1)*(3*n - 5)*a(n-2).

From Peter Bala, Sep 29 2015: (Start)

a(n) = Sum_{i = 0..n} binomial(3*n,i) * binomial(2*n-i-1,n-i).

a(n) = [x^n] ( (1 + x)^3/(1 - x) )^n.

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 4*x + 23*x^2 + 156*x^3 + 1162*x^4 + 9192*x^5 + ... is the o.g.f. for A007297 (but with an offset of 0). (End)

a(n) = (n+1)*A078531(n). [Barry, JIS (2011)]

G.f.: x*B'(x)/B(x), where x*B(x)+1 is g.f. of A007297. - Vladimir Kruchinin, Oct 02 2015

From Peter Bala, Aug 22 2016: (Start)

a(n) = Sum_{k = 0..floor(n/2)} binomial(4*n,n-2*k)*binomial(n+k-1,k).

O.g.f.: A(x) = Hypergeom([5/6, 1/6], [1/2], 108*x^2) + 4*x*Hypergeom([4/3, 2/3], [3/2], 108*x^2).

The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^3/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197. (End)

a(n) ~ 2^n*3^(3*n/2)/sqrt(2*Pi*n). - Ilya Gutkovskiy, Aug 22 2016

a(n) = 4^n*2*(n+1)*binomial((3*n-1)/2, n+1)/(n-1) for n >= 2. - Peter Luschny, Feb 03 2020

From Peter Bala, Mar 04 2022: (Start)

The o.g.f. A(x) satisfies the algebraic equation (1 - 108*x^2)*A(x)^3 - A(x) = 8*x. Cf. A244039.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.

Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for primes p >= 5 and positive integers n and k. (End)

From Seiichi Manyama, Aug 09 2025: (Start)

a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^n).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k) * binomial(2*n-k,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(2*n-k,n-k).

a(n) = 4^n * binomial((3*n-1)/2,n).

a(n) = [x^n] 1/(1-4*x)^((n+1)/2).

a(n) = [x^n] (1+4*x)^((3*n-1)/2). (End)