Strings over $GF(4)$ 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 $GF(4)$ 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 $n(n-1)/2$ pairs of digits taken over the field, i.e., $s = \sum_{1\leq i\lt j\leq n} a_i a_j$. Below we use $a=\operatorname{root}(x^2+x+1)$ and $b=1+a$.

(trace,subtrace)
$n$ (0,0) (0,1)
(0,$a$)
(0,$b$)
(1,0)
($a$,0)
($b$,0)
(1,1)
($a$,$b$)
($b$,$a$)
(1,$a$)
($a$,1)
($b$,1)
(1,$b$)
($a$,$a$)
($b$,$b$)
1 10 10 0
2 11 22 0
3 73 37 3
4 2812 1616 16
5 7660 7660 60
6 256256 272272 240
7 10721008 10081072 1008
8 42884032 40964096 4096
9 1657616320 1657616320 16320
10 6553665536 6579265792 65280
11 262912261888 261888262912 261888
12 10516481047552 10485761048576 1048576
13 41973764193280 41973764193280 4193280
14 1677721616777216 1678131216781312 16773120
15 6712115267104768 6710476867121152 67104768

Examples

The one 4-ary string of trace 0, subtrace 1 and length 2 is $\{11\}$. The two 4-ary strings of trace $a$, subtrace $b$ and length 2 are $\{1b, b1\}$. The three 4-ary strings of trace $b$, subtrace 1 and length 3 are $\{11b, 1b1, b11\}$.

Enumeration (OEIS)