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A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

%I A358433 #6 Nov 16 2022 14:54:08
%S A358433 2,13,3,365,105,42,43801,12915,6300,2520,21725297,6412815,3228960,
%T A358433 1562400,624960,43798198753,12928608063,6533019360,3254791680,
%U A358433 1574899200,629959680,355991759464385,105083758588095,53109556520832,26576858972160,13227473387520,6400390348800,2560156139520
%N A358433 Triangular array read by rows.  T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.
%C A358433 The index of a matrix A is the smallest positive integer such that rank(A^k) = rank(A^(k+1)).
%e A358433       2,
%e A358433       13,       3,
%e A358433      365,     105,      42,
%e A358433    43801,   12915,    6300,    2520,
%e A358433 21725297, 6412815, 3228960, 1562400, 624960,
%t A358433 nn = 6; 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}];
%t A358433 l[greatestpart_] :=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0,nn}], {2}];g1[u_, v_, deg_] := Total[Map[v ^(If[ Max[Prepend[#, 0]] == 0, 1, Max[Prepend[#, 0]]]) u^(deg Total[#])/aut[deg, #] &, l[nn]]]; Map[Select[#, # > 0 &] &,Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n,0,nn}]CoefficientList[
%t A358433   Series[g1[u, v, 1] g1[u, 1, 1]^(q - 1) Product[g1[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u, v}], 1]] // Grid
%Y A358433 Cf. A002416 (row sums), A348015 (column k=1), A083402 (main diagonal for n>1), A346214.
%K A358433 nonn,tabl
%O A358433 1,1
%A A358433 _Geoffrey Critzer_, Nov 15 2022