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

Author

Stefano Spezia, Jul 04 2023

Keywords

• nonn
• tabl

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.

Examples

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

Links

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

Crossrefs

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

Programs

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