A091510 Number of nonisomorphic algebras with a ternary operation (3-d groupoids) with n elements.

1, 1, 136, 1270933717887, 14178431955039102651224805804387336192, 19591572513704791799478942287037427963655716808579364910828644498251439742675781250000

Offset: 0

Author

Christian G. Bower, Jan 16 2004

Keywords

• nonn

Links

• Philip Turecek, Table of n, a(n) for n = 0..10

Crossrefs

Cf. A001329, A001331, A091511.

Programs

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

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = Product_{i, j, k>=1} ( (Sum_{d|lcm(i, j, k)} (d*s_d))^(s_i*s_j*s_k*lcm(i, j, k)/(i*j*k))).

a(n) is asymptotic to n^(n^3)/n!.