Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Aug 24 2021 06:49:06
%S 1,0,2,0,2,14,0,2,98,412,0,2,1542,13160,50832,0,2,34782,1147744,
%T 6854720,25517184,0,2,1908734,260411904,2544075264,14153094144,
%U 51759986688,0,2,166738046,107691724672,2985421682688,21570911944704,116285097148416,422000664182784
%N Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with minimal polynomial of degree k, n >= 0, 0 <= k <= n.
%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%e 1,
%e 0, 2,
%e 0, 2, 14,
%e 0, 2, 98, 412,
%e 0, 2, 1542, 13160, 50832
%t nn = 8; q = 2; b[p_, i_] := Count[p, i];d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] =
%t Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; L = Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; g[u_, v_, deg_] :=
%t Total[Map[v^(Max[Prepend[#, 0]] deg) u^(deg Total[#])/aut[deg, #] &,
%t L]]; Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[g[u, v, deg]^\[Nu][[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid
%Y Cf. A002416 (row sums), A346082 (main diagonal).
%K nonn,tabl
%O 0,3
%A _Geoffrey Critzer_, Aug 10 2021