TITLE: Exact formula for $\chi(X, \, S^n \Omega^1_X)$
QUESTION [7 upvotes]: I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1_X),$$
where $S^n$ denotes the symmetric product, in terms of $c_1(X), c_2(X)$.
I know how to do the computation, by using the splitting principle in order to calculate the Chern classes $c_i(S^n \Omega^1_X)$ and then the Hirzebruch-Riemann-Roch formula, and I did it for small values of $n$. However, extracting a formula for general $n$ seems quite tedious.

Question. Is there a reference where I can find the value of  $\chi(X, \, S^n \Omega^1_X)$? Related to this: is there a reference
for the Chern classes $c_1(S^n \Omega^1_X)$, $c_2(S^n \Omega^1_X)$?

Notice that I need an exact formula, not an asymptotic one (which I already know).

REPLY [4 votes]: Just for reference: using the Schubert2 package from Macaulay2 this can be quite effortlessly done.
i1 : needsPackage "Schubert2";

i2 : X = abstractVariety(2, QQ[n,c1,c2,Degrees=>{0,1,2}]);

i3 : T = abstractSheaf(X, Rank=>2, ChernClass=>1+c1+c2);

i4 : X.TangentBundle = T;

i5 : chi symmetricPower(n, cotangentBundle X)

              1 3  2   1 3      1    2   1 2      1  2   1        1
o5 = integral(-n c1  - -n c2 - --n*c1  - -n c2 + --c1  - -n*c2 + --c2)
              6        6       12        2       12      4       12

If one looks at the implementation, the $n$-th symmetric power of a bundle $\mathcal F$ is computed using the pushforward of the relative $\mathcal{O}(n)$ on the projectivization $\mathbf{P}(\mathcal F^\vee)$ (as already mentioned in the other answer).