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

0,3

Links

Table of n, a(n) for n=0..30.

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

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).

Example

  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;

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

Crossrefs

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

Sequence in context: A186634 A363910 A139213 * A306079 A373167 A242860

Adjacent sequences: A344870 A344871 A344872 * A344874 A344875 A344876

Keyword

nonn,tabf

Author

Geoffrey Critzer, Jul 12 2021

Status

approved