A389829 a(n) is the number of closed binary operations on n elements, labeled, that satisfy both x(yz) = (xy)z and x(yz) = xz identically.

1, 1, 4, 17, 84, 507, 3668, 31117, 312938, 3669671, 48071832, 690121137, 10912283462, 189568637611, 3551946948020, 70691032612277, 1495186187390130, 33904761111198159, 825492020819021552, 21358930280058590041, 580041780169963055582

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

David Pasino, Semigroups satisfying also X(YZ) = XZ identically.

Formula

a(n) = Sum_{k=1..n} Sum_{r*s=k} binomial(n,k)*k^(n-k)*k!/(r!*s!) for n > 0.

Prog

(PARI) a(n) = if(n==0, 1, sum(k=1, n, sumdiv(k, d, binomial(n, k)*k^(n-k)*k!/(d!*(k/d)!) ))) \\ Andrew Howroyd, Nov 14 2025

Crossrefs

Cf. A023814, A121860, A279644, A390162 (unlabeled case).

Sequence in context: A200716 A093904 A093344 * A087316 A104979 A081052

Adjacent sequences: A389826 A389827 A389828 * A389830 A389831 A389832

Keyword

nonn

Author

David Pasino, Nov 14 2025

Status

approved