A029850 Number of self-converse groupoids.

1, 1, 4, 138, 60160, 453292525, 72471180989664, 298545867396801815077, 37263960166680610905649057368, 161614516495439236943507628117344255307, 27480138271604938271870114918720067827110789528890

Offset: 0

Author

Christian G. Bower, Jan 15 1998, Jun 15 1998, Dec 05 2003

Keywords

• nonn

Links

• Index entries for sequences related to groupoids

Crossrefs

a(n) = 2*A001424(n) - A001329(n). Cf. A001425.

Formula

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*2} (d*s_d))^((i*s_i^2-s_i)/2) * (sum {d|i} (d*s_d))^s_i or {i=j == 0 mod 4} (sum {d|i} (d*s_d))^(i*s_i^2) or {i=j == 2 mod 4} (sum {d|i} (d*s_d))^(i*s_i^2-s_i) * (sum {d|i/2} (d*s_d))^(2*s_i) or {i != j} (sum {d|lcm(i, j, 2)} (d*s_d))^(2*i*j*s_i*s_j/lcm(2*i*j)).

Extensions

Formula corrected by Sean A. Irvine and Christian G. Bower, Jul 13 2012