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.

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, 108, 410, 414, 91, 1, 188, 762, 848, 242, 8, 352, 1390, 1719, 601, 34, 632, 2616, 3406, 1416, 122, 1182, 4879, 6739, 3207, 374, 3, 2192, 9196, 13274, 7026, 1062, 18

Offset: 1

Author

Geoffrey Critzer, Nov 30 2019

Keywords

• nonn
• tabf

Comments

Observed row lengths are 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, ...

Examples

			    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;
  108, 410, 414, 91, 1;
  ...
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).
		

Crossrefs

Cf. A306945, A269456.

Row sums give A000079.

Column k=1 gives A000031.

Programs

Mathematica

nn = 10; a = Table[1/m Sum[MoebiusMu[m/d] 2^d, {d, Divisors[m]}], {m, 1,
   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]]]

Formula

G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001037(k).