Triangular array read by rows. T(n,k) is the number of degree n polynomials p in GF_2[x] such that the square free part of p has degree k, n >= 0, 0 <= k <= n.

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

1,

1;

0, 2,;

2, 0, 2,;

2, 2, 0, 4,;

4, 2, 2, 0, 8,;

4, 4, 2, 6, 0, 16,;

10, 2, 4, 6, 10, 0, 32,;

8, 10, 4, 10, 10, 22, 0, 64;

proposed

editing

editing

proposed

Triangular array read by rows. T(n,k) is the number of degree n polynomials p in GF_2[x] such that the square free part of p has degree k, n>=0, 0<=k<=n.

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

1,

allocated for Geoffrey CritzerTriangular array read by rows. T(n,k) is the number of degree n polynomials p in GF_2[x] such that the square free part of p has degree k, n.

1, 0, 2, 2, 0, 2, 2, 2, 0, 4, 4, 2, 2, 0, 8, 4, 4, 2, 6, 0, 16, 10, 2, 4, 6, 10, 0, 32, 8, 10, 4, 10, 10, 22, 0, 64, 20, 4, 10, 10, 20, 22, 42, 0, 128, 20, 18, 6, 24, 16, 44, 42, 86, 0, 256, 40, 14, 18, 18, 48, 38, 80, 86, 170, 0, 512, 40, 36, 16, 48, 32, 106, 68, 166, 170, 342, 0, 1024

0,3

1,

0, 2,

2, 0, 2,

2, 2, 0, 4,

4, 2, 2, 0, 8,

4, 4, 2, 6, 0, 16,

10, 2, 4, 6, 10, 0, 32,

8, 10, 4, 10, 10, 22, 0, 64

nn = 12; q = 2; a = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; Table[Take[CoefficientList[ Series[Product[(1/(1 - z^i) - z^i + u^i z^i)^a[[i]], {i, 1, nn}], {z, 0, nn}], {z, u}][[j]], j], {j, 1, nn}] // Grid

allocated

nonn,tabl

Geoffrey Critzer, Aug 13 2022

approved

editing

allocated for Geoffrey Critzer

allocated

approved