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Row 2 of square array A361027.

The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result 60*(2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that 20160*A006480(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^2.

Equivalently, the central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that (1*3*5)*A000984(n+3) is divisible by (2*n + 1)*(2*n + 3)*(2*n + 5). The result is always an even integer. In fact, (1/2)*(1*3*5)/((2*n + 1)*(2n + 3)*(2n + 5))*A000984(n+3) = A007272(n).

Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that (1*2*4*5*7*8)*A006480(n+3) is divisible by (3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8). The result is always an integer divisible by 3.

Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!).

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.

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).

The Catalan numbers A000108 are defined by the formula Catalan(n) = (2*n)!/(n!*(n+1)!). Gessel (1992) considered generalized Catalan numbers defined by Catalan(r,n) = J(r)*(2*n)!/(n!*(n+r+1)!), where J(r) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. Gessel's generalized Catalan numbers are particular cases of super ballot numbers. See A135573 for a table of these generalized Catalan numbers.

For r = 0,1,2,..., it appears that there is an integer C(r) such the sequence {C(r)*(3*n)!/(n!*(2*n + r)!) : n >= 0} is integral. This is the case r = 3. For other cases see A005809 (r = 0, C(0) = 1), A001764 (r = 1, C(1) = 1), A000139 (r = 2, C(2) = 4) and A361041 (r = 4, C(4) = 1680).