Strings over $GF(2)$ with given trace and subtrace
Here we consider the number $S(n;t,s)$ of length $n$ words $a_1 a_2 \ldots a_n$ over the alphabet consisting of the elements of the field $GF(2)$ that have trace $t$ and subtrace $s$.
The
trace of a word is the sum of its digits over the field, i.e., $t = a_1 + a_2 + \cdots + a_n$.
The
subtrace is the sum of the products of all $n(n1)/2$ pairs of digits taken over the field, i.e., $s = \sum_{1\leq i\lt j\leq n} a_i a_j$.
Note that $GF(2) = \mathbb{Z}_2$.
 (trace,subtrace) 
$n$
 (0,0)
 (0,1)
 (1,0)
 (1,1)

1 
1  0  1
 0


2 
1  1  2
 0


3 
1  3  3
 1


4 
2  6  4
 4


5 
6  10  6
 10


6 
16  16  12
 20


7 
36  28  28
 36


8 
72  56  64
 64


9 
136  120  136
 120


10 
256  256  272
 240


11 
496  528  528
 496


12 
992  1056  1024
 1024


13 
2016  2080  2016
 2080


14 
4096  4096  4032
 4160


15 
8256  8128  8128
 8256


16 
16512  16256  16384
 16384


17 
32896  32640  32896
 32640


18 
65536  65536  65792
 65280


19 
130816  131328  131328
 130816


20 
261632  262656  262144
 262144


Examples
The two binary strings of trace 1, subtrace 0 and length 2 are $\{10, 01\}$.
The three binary strings of trace 0, subtrace 1 and length 3 are $\{ 011, 101, 110\}$.
The four binary strings of trace 1, subtrace 1 and length 4 are $\{0111, 1011, 1101, 1110\}$.
Enumeration (OEIS)

The number $S(n;t,s)$ can be computed from the following recurrence relation
\begin{align}
S(n;t,s) &= S(n1;t,s) + S(n1;t1,s(t1)) \\
&= S(n1;t,s) + S(n1;t+1,s+t+1).
\end{align}
Note that all operations involving operands $t$ or $s$ are carried out over $GF(2)$.

Column (0,0) is OEIS A038503.

Column (0,1) is OEIS A038505.

Column (1,0) is OEIS A038504.

Column (1,1) is OEIS A000749.