OFFSET
0,2
COMMENTS
Equivalently, a(n) is the number of n X n matrices over GF(2) whose invariant factors are the same as its elementary divisors. In other words, the matrices whose rational canonical form is equal to the primary rational canonical form.
LINKS
MATHEMATICA
nn = 12; q = 2; b[p_, i_] := Count[p, i]; s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], n, 1, nn}]; l[greatestpart_] := Level[Table[ IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}]; g[u_, v_, deg_] := Total[Map[v^Total[#] u^(deg Total[#])/aut[deg, #] &, l[nn]]]; Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
Series[1 + Sum[\[Nu][[d]]*(g[u, 1, d] - 1), {d, 1, nn}] , {u, 0, nn}], u]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 02 2022
STATUS
approved