For even numbered periods the centered moving average is calculated incorrectly. Suppose the period length is 5. Then the center of 5 periods is 3. However if the period length is 4 then the center of the period is 2.5. The value at index 3 should be the average of the values at 2.5 and 3.5. Pandas is showing the 2.5 value at 3 which is incorrect.

EXAMPLE:
a=pd.Series(range(1,6), index=range(1,6))
a.rolling(4, center=True).mean()

1 NaN
2 NaN
3 2.5
4 3.5
5 NaN