A364071 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*8^(n-d-k), with 0 <= k <= n.

1, 1, 1, 1, 10, 1, 1, 91, 27, 1, 1, 820, 550, 52, 1, 1, 7381, 10170, 1850, 85, 1, 1, 66430, 180271, 56420, 4655, 126, 1, 1, 597871, 3131037, 1590771, 210035, 9821, 175, 1, 1, 5380840, 53825500, 42900312, 8521926, 612696, 18396, 232, 1, 1, 48427561, 920414340, 1126333300, 324123870, 33642462, 1514100, 31620, 297, 1

Offset: 0

Author

Stefano Spezia, Jul 04 2023

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of 8-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,     10,       1;
  1,     91,      27,      1;
  1,    820,     550,     52,       1;
  1,   7381,   10170,   1850,      85,    1;
  1,  66430,  180271,  56420,    4655,  126,   1;
  1, 597871, 3131037, 1590771, 210035, 9821, 175, 1;
  ...
		

Links

• Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.

Crossrefs

Cf. A000012 (k=0), A002452 (k=1), A003580 (row sums), A364072, A364073.

Programs

Mathematica

T[n_,k_]:=Sum[Binomial[n,d]StirlingS2[n-d,k]8^(n-d-k),{d,0,n-k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten