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a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,n-3*k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 27*(n-2)*(n-1)^2*a(n-3) + 108*(n-2)*(n^2 - 3*n + 1)*a(n-4) + 54*(3*n^3 - 18*n^2 + 28*n - 5)*a(n-5) + 108*(n^3 - 7*n^2 + 12*n - 1)*a(n-6) + 27*(n-5)*(n-3)*n*a(n-7).

a(n) ~ sqrt(3) * ((3 + sqrt(21))/2)^n / (2*Pi*n). (End)

a(n) = Sum_{k=0..n} (3*k)!/k!^3 * binomial(3*k,n-k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: (n-1)*n^2*a(n) = 2*(n-1)*(13*n^2 - 13*n + 3)*a(n-1) + 12*(9*n^3 - 27*n^2 + 23*n - 3)*a(n-2) + 18*(9*n^3 - 36*n^2 + 38*n - 3)*a(n-3) + 12*(9*n^3 - 45*n^2 + 56*n - 2)*a(n-4) + 3*n*(3*n - 11)*(3*n - 7)*a(n-5).

a(n) ~ c * d^n / n, where d = 29.8094342438507627973286122946283855557156321402886102401458498265933891... is the real root of the equation -27 - 81*d - 81*d^2 - 27*d^3 + d^4 = 0 and c = sqrt(3)/(2*Pi) = 0.27566444771089602475566324915648472069869324018332... (End)

a(n) = Sum_{k=0..floor(n/2)} (3*k)!/k!^3 * binomial(k,n-2*k).

From Vaclav Kotesovec, Mar 23 2023: (Start)

Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 3*(n-1)*(3*n - 4)*(3*n - 2)*a(n-2) + 18*(n-2)*(3*n^2 - 6*n + 1)*a(n-3) + 27*(n-3)*(n-2)*n*a(n-4).

a(n) ~ sqrt(3) * (6*cos(Pi/9))^n / (2*Pi*n). (End)