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Sage

Pol. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n==0: return Pol.one()
    return sum(x[k]*Z(n-k) for k in (1..n))/n
def a(n,k=3):
    P = Z(n)
    q = 0
    coeffs = P.coefficients()
    for mon in enumerate(P.monomials()):
        m = Pol(mon[1])
        p = 1
        V = m.variables()
        T = cartesian_product(k*[V])
        Tsorted = [tuple(sorted(u)) for u in T]
        Tset = set(Tsorted)
        for t in Tset:
            r = [Pol.varname_key(str(u))[1] for u in t]
            j = [m.degree(u) for u in t]
            D = 0
            lcm_r = lcm(r)
            for d in divisors(lcm_r):
                try: D += d*m.degrees()[-d-1]
                except: break
            p *= D^(Tsorted.count(t)*prod(r)/lcm_r*prod(j))
        q += coeffs[mon[0]]*p
    return q
# Philip Turecek, Jun 12 2023