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A361040 a(n) = 420*(3*n)!/(n!*(2*n + 3)!).

%I A361040 #12 Jul 28 2024 03:44:13
%S A361040 70,21,30,70,210,735,2856,11970,53130,246675,1187550,5890248,29954680,
%T A361040 155602020,823184880,4424618730,24116031162,133072694475,742405558650,
%U A361040 4182821562150,23776769743650,136248095712855,786482994679200
%N A361040 a(n) = 420*(3*n)!/(n!*(2*n + 3)!).
%C A361040 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.
%C A361040 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).
%H A361040 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 A361040 a(n) = 70*binomial(3*n,2*n) - 189*binomial(3*n,2*n+1) + 114*binomial(3*n,2*n+2) -32*binomial(3*n,2*n+3). Thus a(n) is an integer.
%F A361040 P-recursive: 2*(n + 1)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70.
%F A361040 a(n) ~ (27/4)^n * 105*sqrt(3/(16*Pi))/n^(7/2).
%F A361040 The o.g.f. A(x) satisfies the differential equation
%F A361040 x^2*(4 - 27*x^4)*A''(x) + 2*x*(7 - 27*x)*A'(x) + (6 - 6*x)*A(x) - 420 = 0, with A(0) = 70 and A'(0) = 21.
%p A361040 seq( 420*(3*n)!/(n!*(2*n + 3)!), n = 0..20)
%Y A361040 Cf. A000139, A001764, A005809, A007272, A135573, A361038, A361041.
%K A361040 nonn,easy
%O A361040 0,1
%A A361040 _Peter Bala_, Mar 04 2023