Is it possible to get a symbolic a concise expression for my function?

Question

Suppose I have ordered rank-4 indices $1\le i<j<k<l\le n$, where $n$ is a fixed but unspecified integer. I define a function $s(n)$ as follows:

$$s(n)=\sum_{1\le i<j<k<l\le n}(-1)^{\text{number of elements in the intersection }\{i,j,k,l\}\cap\{1,2,3,4\}}.$$

I'm able to implement the above definition in Mathematica:

Remove[a, c, s];
a[n_Integer] :=
  a[n] = 
    Flatten[
      Table[F[i, j, k, l], {l, 1, n}, {k, 1, l - 1}, {j, 1, k - 1}, {i, 1, j - 1}]] 
        /. F[i_, j_, k_, l_] -> {i, j, k, l};
c = {1,2,3,4};
s[n_Integer] := 
  Sum[(-1)^Length[Intersection[a[n][[u]], c]], {u, 1, Length[a[n]]}];

This is giving me correct results if I substitute numerics into it, such as s[5] = -3, s[6] = -1, etc. But I want Mathematica to give me a simplified symbolic result for s[n] in terms of elementary operations like sum, power, factorial, etc. Is there a way to achieve this?

Answer 1

One possible solution I found is to use InterpolatingPolynomial, since we know the result must be a polynomial in n:

Remove[a, c, s];
   a[n_Integer] := a[n] = List @@@ Flatten[Table[F[i, j, k, l], 
{l, 1, n}, {k, 1, l - 1}, {j, 1, k - 1}, {i, 1, 
   j - 1}]];
c = {1, 2, 3, 4};
s[n_Integer] := 
Sum[  (-1)^Length[Intersection[a[n][[u]], c]], {u, 1, Length[a[n]]}];
pts = Table[{n, s[n]}, {n, 5, 9}];
Expand[InterpolatingPolynomial[pts, x]]

which gives the correct result

192 - (1283 x)/12 + (491 x^2)/24 - (19 x^3)/12 + x^4/24