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A344873 Irregular triangle read by rows. T(n,k) is the number of n X n matrices over GF(2) whose characteristic polynomial is a product of k distinct squarefree irreducible factors.

%I A344873 #33 Mar 22 2025 04:40:03
%S A344873 1,0,2,0,2,6,0,48,112,0,4032,11520,6720,0,1935360,4952064,2856960,0,
%T A344873 2879815680,9558687744,7871496192,0,23222833643520,66748107718656,
%U A344873 60247322394624,15604761231360,0,629183972848435200,2137709262359494656,2101670528396820480,465681743169454080
%N A344873 Irregular triangle read by rows. T(n,k) is the number of n X n matrices over GF(2) whose characteristic polynomial is a product of k distinct squarefree irreducible factors.
%H A344873 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.
%F A344873 Sum_{n>=0} Sum_{k>=0} T(n,k)*y^k*x^n/A002884(n) = Product_{d>=1} (1 + y*x^d/(2^d-1))^A001037(d).
%e A344873 Triangle begins:
%e A344873   1;
%e A344873   0,              2;
%e A344873   0,              2,              6;
%e A344873   0,             48,            112;
%e A344873   0,           4032,          11520,           6720;
%e A344873   0,        1935360,        4952064,        2856960;
%e A344873   0,     2879815680,     9558687744,     7871496192;
%e A344873   0, 23222833643520, 66748107718656, 60247322394624, 15604761231360;
%t A344873 nn = 8; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}];Prepend[Drop[Map[Prepend[#, 0] &,Map[Select[#, # > 0 &] &,Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0,nn}] CoefficientList[Series[Product[(1 + v u^i/(2^i - 1))^A001037[[i]], {i, 1, nn}], {u, 0, nn}], {u, v}]]], 1], {1}] // Grid
%Y A344873 Cf. A002884, A001037, A345463 (column k=1), A346164 (row sums).
%K A344873 nonn,tabf
%O A344873 0,3
%A A344873 _Geoffrey Critzer_, Jul 12 2021