How many ways can you roll an even number sum if you have the choice to use up to 5 dice?
even number sums will be 1/2 the total sums for a given number of dice
There are $6^k$ total possible combos of $k$ dice. Half these combos will sum to an even number.
So we have
$n = 3 + 18 + 108 + 648 + 3888 = 4665$ total ways to roll an even sum of up to 5 dice.
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Look at the expansion of the generating function: HERE .
The sum of the coefficients of terms with even exponents is:
$1+15+70+210+470+826+1190+1420+1190+826+270+70+15+ 1=6574$
Please note that these sums include using zero to five dice on any toss.
If you must use five dice then look HERE Sum the coefficients of terms with even exponents.
If you need only "at least one die must be used". Look HERE