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A329721 Irregular 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 distinct factors in its unique factorization into irreducible polynomials.

%I A329721 #33 Dec 05 2019 19:10:02
%S A329721 2,3,1,4,4,6,9,1,8,20,4,14,35,15,20,70,36,2,36,122,90,8,60,226,196,30,
%T A329721 108,410,414,91,1,188,762,848,242,8,352,1390,1719,601,34,632,2616,
%U A329721 3406,1416,122,1182,4879,6739,3207,374,3,2192,9196,13274,7026,1062,18
%N A329721 Irregular 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 distinct factors in its unique factorization into irreducible polynomials.
%C A329721 Observed row lengths are 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, ...
%F A329721 G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001037(k).
%e A329721     2;
%e A329721     3,   1;
%e A329721     4,   4;
%e A329721     6,   9,   1;
%e A329721     8,  20,   4;
%e A329721    14,  35,  15;
%e A329721    20,  70,  36,  2;
%e A329721    36, 122,  90,  8;
%e A329721    60, 226, 196, 30;
%e A329721   108, 410, 414, 91, 1;
%e A329721   ...
%e A329721 T(5,3) = 4 because we have: x(x+1)(x^3+x+1), x(x+1)(x^3 +x^2+1), x^2(x+1)(x^2+x+1), x(x+1)^2(x^2+x+1).
%t A329721 nn = 10; a = Table[1/m Sum[MoebiusMu[m/d] 2^d, {d, Divisors[m]}], {m, 1,
%t A329721    nn}]; Grid[Map[Select[#, # > 0 &] &, Drop[CoefficientList[Series[Product[(u/(1 -  z^m ) - u + 1)^a[[m]], {m, 1, nn}], {z, 0,nn}], {z, u}], 1]]]
%Y A329721 Cf. A306945, A269456.
%Y A329721 Row sums give A000079.
%Y A329721 Column k=1 gives A000031.
%K A329721 nonn,tabf
%O A329721 1,1
%A A329721 _Geoffrey Critzer_, Nov 30 2019