OFFSET
0,2
FORMULA
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)
MATHEMATICA
Table[Sum[(3*k)!/k!^3 * Binomial[3*k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2023 *)
PROG
(PARI) a(n) = sum(k=0, n, (3*k)!/k!^3*binomial(3*k, n-k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 22 2023
STATUS
approved