What is sum of degrees of the irreducible complex characters of the alternating groups?


The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra of the group algebra $\mathbb{C} A\_n$ for all $n\in \mathbb{N}$.


The corresponding sum for the symmetric group is known by the Frobenius-Schur-Indicators that is to calculate the involutions plus 1 in the symmetric group. This value is also an upper bound for the sum related to the alternating group (see also <https://math.stackexchange.com/questions/590402/number-of-involution-in-symmetric-group>)


I would like to know the difference betweeen these two values.


In addition the character theory of the alternating groups are closely related to the symmetric group by using Clifford-Theory: some irreducible are still irreducible by restricting them to the alternative group the other ones decompose into two irreducible characters.


In the article <http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JACO/Volume5_2/lmm440901p5p3660.fulltext.pdf> these two irreducible characters of $A\_n$ are investigated based on theorems by Thrall.



Sum of irreducible complex character degrees for alternating groups
For small $n$, you can compute the sums of irreducible complex character degrees of ${\rm A}\_n$ with [GAP](http://www.gap-system.org):



```
gap> sums := List([3..15],n->Sum(List(CharacterDegrees(AlternatingGroup(n)),Product)));
[ 3, 6, 16, 46, 126, 448, 1366, 5356, 18568, 76296, 297012, 1264264, 5412928 ]

```

However I doubt there is a nice formula for these values.


