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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.
0
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
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
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Jul 12 2021
STATUS
approved