A356583 T(n,k) is the number of degree n polynomials p in GF_2[x] whose squarefree part has degree k, n >= 0, 0 <= k <= n. Triangular array read by rows.

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

Offset

0,3

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

   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;

Mathematica

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

Crossrefs

Cf. A001037.

Sequence in context: A354186 A366265 A127527 * A217943 A177225 A236306

Adjacent sequences: A356580 A356581 A356582 * A356584 A356585 A356586

Keyword

nonn,tabl

Author

Geoffrey Critzer, Aug 13 2022

Status

approved