%I #11 Dec 19 2021 13:55:39
%S 1,6,996,31857648,266666713602640,929809173755713574913480,
%T 2002123402266181527640478418179038176,
%U 3702236248557739850415303240942330019881771301360640
%N Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.
%C A079187(n)+A079190(n)=A001329(n).
%C Each a(n) is equal to the sum of the elements in row n of A079191.
%H C. van den Bosch, <a href="http://cosmos.ucc.ie/~cjvdb1/html/binops.shtml">Closed binary operations on small sets</a>
%H <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</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*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (sum {d|i} (d*s_d))^(s_i*(i*s_i+1)/2) * (-1 + sum {d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (sum {d|i and d/i is odd} (d*s_d))^s_i * (sum {d|i} (d*s_d))^(i*s_i^2/2) * (-1 + sum {d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)
%F a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)
%Y Cf. A079187, A079189, A079191.
%K nonn
%O 1,2
%A Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003
%E Edited, corrected and extended with formula by _Christian G. Bower_, Dec 12 2003