A361037 - OEIS

%I #18 Aug 05 2024 15:35:09

%S 10,10,25,84,330,1430,6630,32300,163438,852150,4552275,24812400,

%T 137547000,773564328,4405019090,25357898940,147375745990,863805209750,

%U 5101386767295,30332569967700,181465130121450,1091677288630950

%N a(n) = 20*(3*n)!/((2*n)!*(n+2)!).

%C Gessel (1992) introduced sequences {b(r,n): n >= 0} of super ballot numbers defined by b(r,n) = J(r) * (2*n)!/(n!*(n + r + 1)!), r = 0,1,2,..., where J(r) = (2*r + 2)!/(2*(r + 1)!) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. The sequence {b(1,n) : n >= 0} is A000108, the sequence of Catalan numbers. See A135573 for a table of these generalized Catalan numbers.

%C We carry out an analogous construction using the numbers B(n) = A005809(n) = binomial(3*n,n) = (3*n)!/((2*n)!*n!) in place of the central binomial numbers. We define B(r,n), r = 0,1,2, ..., by B(r,n) = F(r) * (3*n)!/((2*n)!*(n + r + 1)!), where F(r) is the minimal choice to produce integer values for these quantities for all n. This sequence is the case r = 1. See A007226 (r = 0), A361038 (r = 2) and A361039 (r = 3).

%H Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>, J. Symbolic Comp., 14 (1992), 179-194.

%F a(n) = 10*binomial(3*n,n) - 7*binomial(3*n,n+1) + binomial(3*n,n+2). Thus a(n) is an integer.

%F P-recursive: 2*(n + 2)(2*n - 1)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 10.

%F a(n) ~ (27/4)^n * 10*sqrt(3/Pi)/n^(5/2).

%F The o.g.f. satisfies the differential equation

%F x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 5)*A'(x) + 2*(3*x + 2)*A(x) - 40 = 0, with A(0) = 10 and A'(0) = 10.

%p seq( 20*(3*n)!/((2*n)!*(n+2)!), n = 0..20);

%t Table[20 (3n)!/((2n)!(n+2)!),{n,0,30}] (* _Harvey P. Dale_, Aug 05 2024 *)

%Y Cf. A000139, A001764, A005809, A007226, A361038, A361039.

%K nonn,easy

%O 0,1

%A _Peter Bala_, Mar 04 2023