%I #23 Jul 19 2023 00:13:47
%S 1,1,136,1270933717887,14178431955039102651224805804387336192,
%T 19591572513704791799478942287037427963655716808579364910828644498251439742675781250000
%N Number of nonisomorphic algebras with a ternary operation (3-d groupoids) with n elements.
%H Philip Turecek, <a href="/A091510/b091510.txt">Table of n, a(n) for n = 0..10</a>
%F 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))).
%F a(n) is asymptotic to n^(n^3)/n!.
%o (Sage)
%o Pol.<x> = InfinitePolynomialRing(QQ)
%o @cached_function
%o def Z(n):
%o if n==0: return Pol.one()
%o return sum(x[k]*Z(n-k) for k in (1..n))/n
%o def a(n,k=3):
%o P = Z(n)
%o q = 0
%o coeffs = P.coefficients()
%o for mon in enumerate(P.monomials()):
%o m = Pol(mon[1])
%o p = 1
%o V = m.variables()
%o T = cartesian_product(k*[V])
%o Tsorted = [tuple(sorted(u)) for u in T]
%o Tset = set(Tsorted)
%o for t in Tset:
%o r = [Pol.varname_key(str(u))[1] for u in t]
%o j = [m.degree(u) for u in t]
%o D = 0
%o lcm_r = lcm(r)
%o for d in divisors(lcm_r):
%o try: D += d*m.degrees()[-d-1]
%o except: break
%o p *= D^(Tsorted.count(t)*prod(r)/lcm_r*prod(j))
%o q += coeffs[mon[0]]*p
%o return q
%o # _Philip Turecek_, Jun 12 2023
%Y Cf. A001329, A001331, A091511.
%K nonn
%O 0,3
%A _Christian G. Bower_, Jan 16 2004