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.

1, 0, 2, 0, 2, 6, 0, 48, 112, 0, 4032, 11520, 6720, 0, 1935360, 4952064, 2856960, 0, 2879815680, 9558687744, 7871496192, 0, 23222833643520, 66748107718656, 60247322394624, 15604761231360, 0, 629183972848435200, 2137709262359494656, 2101670528396820480, 465681743169454080

Offset: 0

Author

Geoffrey Critzer, Jul 12 2021

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  0,              2;
  0,              2,              6;
  0,             48,            112;
  0,           4032,          11520,           6720;
  0,        1935360,        4952064,        2856960;
  0,     2879815680,     9558687744,     7871496192;
  0, 23222833643520, 66748107718656, 60247322394624, 15604761231360;
		

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A002884, A001037, A345463 (column k=1), A346164 (row sums).

Programs

Mathematica

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

Formula

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).