Strings over $GF(5)$ 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 $GF(5)$ 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 $GF(5) = \mathbb{Z}_5$.
 (trace,subtrace) 
$n$
 (0,0)
 (0,1) (0,4)
 (0,2) (0,3)
 (1,0) (2,0) (3,0) (4,0)
 (1,1) (2,4) (3,4) (4,1)
 (1,2) (2,3) (3,3) (4,2)
 (1,3) (2,2) (3,2) (4,3)
 (1,4) (2,1) (3,1) (4,4)

1 
1  0  0
 1
 0  0  0
 0


2 
1  2  0
 2
 0  0  2
 1


3 
1  6  6
 6
 6  1  6
 6


4 
25  20  30
 20
 25  20  30
 30


5 
125  100  150
 125
 125  125  125
 125


6 
625  600  650
 625
 600  650  650
 600


7 
3025  3150  3150
 3150
 3150  3150  3150
 3025


8 
15625  15750  15500
 15500
 15750  15625  15750
 15500


9 
78625  78000  78000
 78000
 78625  78000  78000
 78000


10 
393125  390000  390000
 390625
 390625  390625  390625
 390625


Examples
The two 5ary strings of trace 0, subtrace 4 and length 2 are $\{14, 41\}$.
The two 5ary strings of trace 3, subtrace 2 and length 2 are $\{12, 21\}$.
The one 5ary string of trace 1, subtrace 2 and length 3 is $\{222\}$.
Enumeration (OEIS)

The number $S(n;t,s)$ can be computed from the following recurrence relation
\[
S(n;t,s) = S(n1;t,s) + S(n1;t1,s(t1)) + S(n1;t2,s2(t2)) + S(n1;t3,s3(t3)) + S(n1;t4,s4(t4)).
\]
Note that all operations involving operands $t$ or $s$ are carried out over $GF(5)$.

Column (0,0) is OEIS A073963.

Column (0,1),(0,4) is OEIS A073964.

Column (0,2),(0,3) is OEIS A073965.

Column (1,0),(2,0),(3,0),(4,0) is OEIS A073966.

Column (1,1),(2,4),(3,4),(4,1) is OEIS A073967.

Column (1,2),(2,3),(3,3),(4,2) is OEIS A073968.

Column (1,3),(2,2),(3,2),(4,3) is OEIS A073969.

Column (1,4),(2,1),(3,1),(4,4) is OEIS A073970.