A391093 a(n) is the number of closed binary operations on a set of n labeled elements with a closed center.

1, 1, 16, 16767, 4108582912, 296140458564453125, 10307803328954692112200237056, 256910476646266498327111877219430908525047, 6277080783543967914646628045264727747332736566041721700352, 196627013933507434875288794151851106026746857471839258981120051659247168405385

Offset

0,3

Comments

The center of a closed binary operation is the set of elements x such that xy = yx for every element y.

Every commutative or associative operation has a closed center.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..25

Wikipedia, Center (algebra).

Maple

f := proc(n, k) options operator; n^(((1/2)*n+(1/2)*k+1/2)*(n-k))*k^((1/2)*k*(k+1))*(-1)^(n-k)*add(binomial(n-k, r)*(-1)^r*n^binomial(r, 2), r = 0 .. n-k) end proc:
T := proc(n, k) options operator; binomial(n, k)*f(n, k) end proc:
a := proc(n) options operator; add(T(n, k), k = 0 .. n) end proc:
seq(a(n), n=0..10);

Prog

(PARI)
T(n, k) = {binomial(n, k)*k^binomial(k+1, 2)*sum(j=0, n-k, (-1)^j*binomial(n-k, j)*n^(binomial(j+1, 2)+(n-k)*(n-j)))}
a(n) = {sum(k=0, n, T(n, k))} \\ Andrew Howroyd, Dec 08 2025

Crossrefs

Row sums of A386681.

Cf. A002489, A023814, A391094, A391162 (unlabeled version).

Sequence in context: A017188 A168576 A013831 * A214637 A288957 A159387

Adjacent sequences: A391090 A391091 A391092 * A391094 A391095 A391096

Keyword

nonn

Author

David Pasino, Nov 28 2025

Status

approved