A391482 Triangle read by rows: T(n,k) is the number of labeled commutative magmas with n elements and k idempotents.

1, 0, 1, 2, 4, 2, 216, 324, 162, 27, 331776, 442368, 221184, 49152, 4096, 10000000000, 12500000000, 6250000000, 1562500000, 195312500, 9765625, 7346640384000000, 8815968460800000, 4407984230400000, 1175462461440000, 176319369216000, 14105549537280, 470184984576

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..665 (rows 0..35)

Formula

T(n,k) = binomial(n,k) * n^(n*(n-1)/2) * (n-1)^(n-k).

Example

Triangle begins:
       1;
       0,      1;
       2,      4,      2;
     216,    324,    162,    27;
  331776, 442368, 221184, 49152, 4096;
  ...

Prog

(PARI) T(n, k) = binomial(n, k) * n^(n*(n-1)/2) * (n-1)^(n-k);

Crossrefs

Row sums are A023813.

Main diagonal is A076113.

Column k=0 is A391483.

Cf. A038021 (unlabeled version), A391480.

Sequence in context: A134138 A351427 A381410 * A201911 A048644 A246713

Adjacent sequences: A391479 A391480 A391481 * A391483 A391484 A391485

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 10 2025

Status

approved