A391161 Triangle read by rows: T(n,k) is the number of nonisomorphic magmas with n elements and a closed center of size k.

1, 0, 1, 6, 0, 4, 2463, 252, 0, 129, 168710720, 2360896, 98560, 0, 43968, 2463965471530395, 3853366387000, 18229845000, 474718605, 0, 254429900, 14314557500958748403432016, 1835239210644761036784, 775664270273023776, 1904255124528024, 39627683503488, 0, 30468670170912

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..324 (rows 0..24)

Example

Triangle begins:
          1;
          0,       1;
          6,       0,     4;
       2463,     252,     0, 129;
  168710720, 2360896, 98560,   0, 43968;
  ...

Prog

(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
C(d, r)={sum(i=1, #r, my(t=r[i]); if(d%t==0, t))}
D(u, v, r) = {prod(i=1, #u, prod(j=1, #v, my(g=gcd(u[i], v[j]));  C(u[i]*v[j]/g, r)^g))}
E(v, r) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); C(v[i]*v[j]/g, r)^g)) * prod(i=1, #v, my(t=v[i]); C(t, r)^((t+1)\2)*if(t%2, 1, C(t/2, r)))}
L2(n, pz) = { sum(k=0, n, my(s=0); forpart(p=k, forpart(q=n-k, my(r=concat([pz, p, q])); s += (-1)^#p * permcount(p) * permcount(q) * D(pz, p, r) * D(q, r, r) * E(p, r))); s/(k!*(n-k)!)) }
T(n, k) = { my(s=0); forpart(p=k, s+=permcount(p) * E(p, p) * L2(n-k, p)); s/k! }

Crossrefs

Row sums are A391162.

Main diagonal is A001425.

Column k=0 is A391156.

Cf. A001329, A386681 (labeled version), A391155.

Sequence in context: A196623 A265275 A113024 * A391155 A112280 A204850

Adjacent sequences: A391158 A391159 A391160 * A391162 A391163 A391164

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 11 2025

Status

approved