cp's OEIS Frontend

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

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