A391154 Triangle read by rows: T(n,k) is the number of labeled magmas with n elements whose center contains k elements.

1, 0, 1, 8, 0, 8, 14580, 4374, 0, 729, 4048551936, 226492416, 18874368, 0, 1048576, 295675781250000000, 2312011718750000, 34179687500000, 1220703125000, 0, 30517578125, 10306481396833139345326080000, 7928233330938960936960000, 15078911446429728768000, 87747802561511424000, 1645271298028339200, 0, 21936950640377856

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..350 (rows 0..25)

Wikipedia, Center (algebra).

Formula

T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(n,k) * binomial(n-k,j) * n^(binomial(k+1,2) + binomial(j+1,2) + (n-k)*(n-j)).

T(n,k) = [x^k] Sum_{j=0..n} (x - 1)^j * binomial(n,j) * n^(binomial(j+1, 2) + n*(n-j)).

Example

Triangle begins:
           1;
           0,         1;
           8,         0,        8;
       14580,      4374,        0, 729;
  4048551936, 226492416, 18874368,   0, 1048576;
  ...

Prog

(PARI) T(n, k) = {sum(j=0, n-k, (-1)^j * binomial(n, k) * binomial(n-k, j) * n^(binomial(k+1, 2)+binomial(j+1, 2)+(n-k)*(n-j)))}
(PARI) row(n) = {Vecrev(sum(j=0, n, ('x-1)^j * binomial(n, j) * n^(binomial(j+1, 2) + n*(n-j))))}

Crossrefs

Row sums are A002489.

Main diagonal is A023813.

Column k=0 is A391094.

Cf. A386681, A391155 (unlabeled version).

Sequence in context: A197617 A005076 A386681 * A296182 A019863 A350747

Adjacent sequences: A391151 A391152 A391153 * A391155 A391156 A391157

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 10 2025

Status

approved