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%I #66 Aug 10 2023 17:55:24
%S 1,4,30,256,2310,21504,204204,1966080,19122246,187432960,1848483780,
%T 18320719872,182327718300,1820797698048,18236779032600,
%U 183120225632256,1842826521244230,18581317012684800,187679234340049620,1898554215471513600,19232182592635611060
%N a(n) = ((3*n)!/n!^2)*(Gamma(1+n/2)/Gamma(1+3n/2)).
%C 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
%C 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
%D R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H Michael De Vlieger, <a href="/A091527/b091527.txt">Table of n, a(n) for n = 0..985</a>
%H Peter Bala, <a href="/A276098/a276098.pdf">Some integer ratios of factorials</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.3.
%H Karl Dilcher, Armin Straub, and Christophe Vignat, <a href="https://arxiv.org/abs/1903.11759">Identities for Bernoulli polynomials related to multiple Tornheim zeta functions</a>, arXiv:1903.11759 [math.NT], 2019. See p. 11.
%H I. M. Gessel, <a href="http://arxiv.org/abs/1403.7656">A short proof of the Deutsch-Sagan congruence for connected non crossing graphs</a>, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
%H Bernhard Heim and Markus Neuhauser, <a href="https://arxiv.org/abs/2107.05013">Asymptotic Distribution of the Zeros of recursively defined Non-Orthogonal Polynomials</a>, arXiv:2107.05013 [math.CA], 2021.
%H W. Mlotkowski and K. A. Penson, <a href="http://arxiv.org/abs/1309.0595">Probability distributions with binomial moments</a>, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
%F D-finite with recurrence n*(n - 1)*a(n) = 12*(3*n - 1)*(3*n - 5)*a(n-2).
%F From _Peter Bala_, Sep 29 2015: (Start)
%F a(n) = Sum_{i = 0..n} binomial(3*n,i)*binomial(2*n-i-1,n-i).
%F a(n) = [x^n] ( (1 + x)^3/(1 - x) )^n.
%F 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)
%F a(n) = (n+1)*A078531(n). [Barry, JIS (2011)]
%F G.f.: x*B'(x)/B(x), where x*B(x)+1 is g.f. of A007297. - _Vladimir Kruchinin_, Oct 02 2015
%F From _Peter Bala_, Aug 22 2016: (Start)
%F a(n) = Sum_{k = 0..floor(n/2)} binomial(4*n,n-2*k)*binomial(n+k-1,k).
%F 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).
%F 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)
%F a(n) ~ 2^n*3^(3*n/2)/sqrt(2*Pi*n). - _Ilya Gutkovskiy_, Aug 22 2016
%F a(n) = 4^n*2*(n+1)*binomial((3*n-1)/2, n+1)/(n-1) for n >= 2. - _Peter Luschny_, Feb 03 2020
%F From _Peter Bala_, Mar 04 2022: (Start)
%F The o.g.f. A(x) satisfies the algebraic equation (1 - 108*x^2)*A(x)^3 - A(x) = 8*x. Cf. A244039.
%F 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.
%F 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)
%p a := n -> 4^n * `if`(n<2, 1, (2*(n+1)*binomial((3*n-1)/2, n + 1))/(n-1)):
%p seq(a(n), n=0..18); # _Peter Luschny_, Feb 03 2020
%t Table[((3 n)!/n!^2) Gamma[1 + n/2]/Gamma[1 + 3 n/2], {n, 0, 18}] (* _Michael De Vlieger_, Oct 02 2015 *)
%t Table[4^n Sum[Binomial[k - 1 + (n - 1)/2, k], {k, 0, n}], {n, 0, 18}] (* _Michael De Vlieger_, Aug 28 2016 *)
%o (PARI) a(n)=4^n*sum(i=0,n,binomial(i-1+(n-1)/2,i))
%o (Maxima)
%o 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;
%o taylor(x*diff(B(x),x)/B(x),x,0,10); /* _Vladimir Kruchinin_, Oct 02 2015 */
%o (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
%o (Python)
%o from math import factorial
%o from sympy import factorial2
%o def A091527(n): return int((factorial(3*n)*factorial2(n)<<n)//(factorial(n)**2*factorial2(3*n))) # _Chai Wah Wu_, Aug 10 2023
%Y 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.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Jan 18 2004
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