%I #52 Dec 01 2019 17:07:20

%S 2,1,1,2,2,3,4,1,6,8,2,9,16,7,18,30,14,2,30,60,34,4,56,114,72,14,99,

%T 220,156,36,1,186,422,320,90,6,335,817,671,207,18,630,1564,1364,484,

%U 54,1161,3023,2787,1070,148,3,2182,5818,5624,2362,386,12,4080,11240,11357,5095,947,49

%N Triangular array T(n,k) read by rows: T(n,k) is the number of degree n monic polynomials in GF(2)[x] with exactly k squarefree factors in its unique factorization into irreducible polynomials.

%C T(n,k) is also the number of binary words of length n whose Lyndon factorization is strict, i.e., it contains exactly k factors of distinct Lyndon words.

%H Alois P. Heinz, <a href="/A306945/b306945.txt">Rows n = 1..500, flattened</a>

%F G.f.: Product_{k>=1} (1 + y*x)^A001037(k).

%e Triangular array T(n,k) begins:

%e 2;

%e 1, 1;

%e 2, 2;

%e 3, 4, 1;

%e 6, 8, 2;

%e 9, 16, 7;

%e 18, 30, 14, 2;

%e 30, 60, 34, 4;

%e 56, 114, 72, 14;

%e 99, 220, 156, 36, 1;

%e ...

%p with(numtheory):

%p g:= proc(n) option remember; `if`(n=0, 1,

%p add(mobius(n/d)*2^d, d=divisors(n))/n)

%p end:

%p b:= proc(n, i) option remember; expand(`if`(n=0, x^n, `if`(i<1, 0,

%p add(binomial(g(i), j)*b(n-i*j, i-1)*x^j, j=0..n/i))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=1..20); # _Alois P. Heinz_, May 28 2019

%t nn = 16; a = Table[1/n Sum[2^d MoebiusMu[n/d], {d, Divisors[n]}], {n, 1, nn}]; Map[Select[#, # > 0 &] &, Drop[CoefficientList[

%t Series[Product[ (1 + u z^k)^a[[k]], {k, 1, nn}], {z, 0, nn}], {z, u}], 1]] // Grid

%Y Column k=1 gives A001037.

%Y Row sums give A090129(n+1).

%Y Cf. A269456.

%K nonn,look,tabf

%O 1,1

%A _Geoffrey Critzer_, Mar 25 2019