How many monic polynomials modulo 2 have an odd-degree factor?

Question

Question: How many monic polynomials of degree $d$ over $\mathbb{F}_2$ have a factor of odd degree?

Attempt: When $d$ is odd, there must be an odd-degree factor for any monic polynomial of degree $d$, so we may restrict our consideration to the case when $d$ is even. In this case, I have no idea how to proceed, although I was able to generate the following data using Mathematica:

• $d = 2$: $\#$ of such polynomials = 3
• $d = 4$: $\#$ of such polynomials = 12
• $d = 6$: $\#$ of such polynomials = 51
• $d = 8$: $\#$ of such polynomials = 207
• $d = 10$: $\#$ of such polynomials = 849
• $d = 12$: $\#$ of such polynomials = 3441

Is there a reason why one might expect or not expect to have a closed-form formula for the number of such polynomials?

Edit: In case anyone wants to experiment, I've attached the Mathematica code for the case $d = 2$ below.

AlgebraicDegree[eqn_,vars_List]:=Max[Total[GroebnerBasis`DistributedTermsList[eqn/.Equal:>Subtract,vars][[1,All,1]],{2}]];
AlgebraicDegree[eqn_]:=AlgebraicDegree[eqn,Variables[eqn]];
p=2;
tab=Table[Factor[x^2 + a x + b,Modulus->p],{a,0,p-1},{b,0,p-1}]//Flatten;
counter=0;
For[i=1,i<=Length[tab],i++,list=FactorList[tab[[i]]]//Flatten;If[Length[Select[list,Mod[AlgebraicDegree[#],2]==1&]]>0,counter++];];
counter