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 A361040 a(n) = 420*(3*n)!/(n!*(2*n + 3)!). 2
 70, 21, 30, 70, 210, 735, 2856, 11970, 53130, 246675, 1187550, 5890248, 29954680, 155602020, 823184880, 4424618730, 24116031162, 133072694475, 742405558650, 4182821562150, 23776769743650, 136248095712855, 786482994679200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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). LINKS Table of n, a(n) for n=0..22. Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194. FORMULA 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. P-recursive: 2*(n + 1)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70. a(n) ~ (27/4)^n * 105*sqrt(3/(16*Pi))/n^(7/2). The o.g.f. A(x) satisfies the differential equation 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. MAPLE seq( 420*(3*n)!/(n!*(2*n + 3)!), n = 0..20) CROSSREFS Cf. A000139, A001764, A005809, A007272, A135537, A361038, A361041. Sequence in context: A184886 A361041 A201097 * A033390 A129352 A318571 Adjacent sequences: A361037 A361038 A361039 * A361041 A361042 A361043 KEYWORD nonn,easy AUTHOR Peter Bala, Mar 04 2023 STATUS approved

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Last modified July 21 15:18 EDT 2024. Contains 374474 sequences. (Running on oeis4.)