A364072 - OEIS

%I #6 Jul 06 2023 21:13:54

%S 1,1,1,1,65,1,1,4161,192,1,1,266305,28545,382,1,1,17043521,3891520,

%T 101125,635,1,1,1090785345,511266561,23105270,261780,951,1,1,

%U 69810262081,66021638592,4901267861,89335610,562296,1330,1,1,4467856773185,8454558363265,997262532182,27503177191,267021146,1066366,1772,1

%N 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.

%C 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.

%H Prudence Djagba and Jan Hązła, <a href="https://arxiv.org/abs/2306.16421">Combinatorics of subgroups of Beidleman near-vector spaces</a>, arXiv:2306.16421 [math.RA], 2023. See pp. 7-9.

%e The triangle begins:

%e   1;

%e   1,          1;

%e   1,         65,         1;

%e   1,       4161,       192,        1;

%e   1,     266305,     28545,      382,      1;

%e   1,   17043521,   3891520,   101125,    635,   1;

%e   1, 1090785345, 511266561, 23105270, 261780, 951, 1;

%e   ...

%t 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

%Y Cf. A000012 (k=0), A133853 (k=1), A364069 (row sums), A364071, A364073.

%K nonn,tabl

%O 0,5

%A _Stefano Spezia_, Jul 04 2023