A364072 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k), with 0 <= k <= n.

1, 1, 1, 1, 65, 1, 1, 4161, 192, 1, 1, 266305, 28545, 382, 1, 1, 17043521, 3891520, 101125, 635, 1, 1, 1090785345, 511266561, 23105270, 261780, 951, 1, 1, 69810262081, 66021638592, 4901267861, 89335610, 562296, 1330, 1, 1, 4467856773185, 8454558363265, 997262532182, 27503177191, 267021146, 1066366, 1772, 1

Offset: 0

Author

Stefano Spezia, Jul 04 2023

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of 64-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,         65,         1;
  1,       4161,       192,        1;
  1,     266305,     28545,      382,      1;
  1,   17043521,   3891520,   101125,    635,   1;
  1, 1090785345, 511266561, 23105270, 261780, 951, 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), A133853 (k=1), A364069 (row sums), A364071, A364073.

Programs

Mathematica

T[n_,k_]:=Sum[Binomial[n,d]StirlingS2[n-d,k]63^(n-d-k),{d,0,n-k}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten