Summation bug in 11.2

The problem is not with Sum:

FourierCoefficient[Cos[x], x, -k]
(*  0  *)

FourierCoefficient[Cos[x], x, k]

The second code is also much faster. It suggests to me that FourierCoefficient calls Integrate in the first case and uses a short-cut in the second. In fact Integrate (from a Trace of FourierCoefficient) gives a result that is only generically correct:

Integrate[E^(I k x) Cos[x], {x, -π, π}, 
 Assumptions -> k ∈ Integers, GenerateConditions -> False]
(* -((2 k Sin[k π])/(-1 + k^2))  *)

This is not an answer as why it happens. But the change happened after version 7.

I went back to verion 7 to be able to obtain different result. I tried versions 11, 10, 9, and 8 and they all gave same result as above. But in version 7: