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.
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 04 2023
STATUS
approved