OFFSET
0,1
COMMENTS
Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!).
FORMULA
a(n) = 70*binomial(3*n,2*n) - 196*binomial(3*n,2*n+1) + 141*binomial(3*n,2*n+2) - 65*binomial(3*n,2*n+3) + 14*binomial(3*n,2*n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 2)*(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/(4*Pi))/n^(9/2).
The o.g.f. A(x) satisfies the differential equation
x^2*(4 - 27*x^4)*A''(x) + 2*x*(9 - 27*x)*A'(x) + (12 - 6*x)*A(x) - 840 = 0, with A(0) = 70 and A'(0) = 14.
MAPLE
seq( 1680*(3*n)!/(n!*(2*n + 4)!), n = 0..20);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 04 2023
STATUS
approved