A364073 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k), with 0 <= k <= n.

1, 1, 1, 1, 626, 1, 1, 391251, 1875, 1, 1, 244531876, 2733126, 3748, 1, 1, 152832422501, 3658206250, 9753130, 6245, 1, 1, 95520264063126, 4721932028751, 21925818740, 25346895, 9366, 1, 1, 59700165039453751, 5993213367973125, 45788990528771, 85217015555, 54578181, 13111, 1

Offset

0,5

Comments

T(n, k) is the number of 625-subgroups of R^n which have dimension k, where R^n is a near-vector space over a proper nearfield R.

Links

Table of n, a(n) for n=0..35.

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-9.

Example

The triangle begins:
  1;
  1,            1;
  1,          626,          1;
  1,       391251,       1875,       1;
  1,    244531876,    2733126,    3748,    1;
  1, 152832422501, 3658206250, 9753130, 6245, 1;
  ...

Mathematica

T[n_, k_]:=Sum[Binomial[n, d]StirlingS2[n-d, k]624^(n-d-k), {d, 0, n-k}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}]//Flatten

Crossrefs

Cf. A000012 (k=0), A364070 (row sums), A364071, A364072.

Sequence in context: A120320 A013837 A050448 * A045171 A345513 A345766

Adjacent sequences: A364070 A364071 A364072 * A364074 A364075 A364076

Keyword

nonn,tabl

Author

Stefano Spezia, Jul 04 2023

Status

approved