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%I #17 Jul 21 2021 09:28:38
%S 1,1,3,1,21,42,1,315,1260,2520,1,6975,104160,312480,624960,1,373023,
%T 23436000,104993280,314979840,629959680,1,32252031,9175162752,
%U 121912197120,426692689920,1280078069760,2560156139520
%N Triangular array read by rows. T(n,k) is the number of nilpotent n X n matrices over GF(2) with index k, 1 <= k <= n, n >= 1.
%C The index of a nilpotent matrix A is the smallest positive integer k such that A^k = 0.
%e 1,
%e 1, 3,
%e 1, 21, 42,
%e 1, 315, 1260, 2520,
%e 1, 6975, 104160, 312480, 624960
%t 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]}] /. q -> 2; gl2 =
%t Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, 50}]; Table[Table[
%t Sum[gl2[[n]]/ aut[1, Select[IntegerPartitions[n], #[[1]] == k &][[i]]], {i, 1,
%t Length[Select[IntegerPartitions[n], #[[1]] == k &]]}], {k, 1, n}], {n, 1, 7}] // Grid
%Y Cf. A083402 (main diagonal), A053763 (row sums).
%K nonn,tabl,more
%O 1,3
%A _Geoffrey Critzer_, Jul 10 2021