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

1,1

Comments

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

Links

Table of n, a(n) for n=1..58.

Formula

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

Example

    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).

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]]]

Crossrefs

Cf. A306945, A269456.

Row sums give A000079.

Column k=1 gives A000031.

Sequence in context: A298637 A034867 A323893 * A193790 A055446 A104706

Adjacent sequences: A329718 A329719 A329720 * A329722 A329723 A329724

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 30 2019

Status

approved