how to get a rolling mean with mean from previous window

I am desperatly searching for a solution with pandas. Maybe you could help me.

I am looking for a rolling mean with consideration of the previous mean.

df looks like this:

index	count
0	4
1	6
2	10
3	12

now, using the rolling(window=2).mean() function I would get something like this:

index	count	r_mean
0	4	NaN
1	6	5
2	10	8
3	12	11

I would like to consider the mean from the first calculation, like this:

index	count	r_mean
0	4	NaN
1	6	5
2	10	7.5
3	12	9.5

where,

row1: (4+6)/2=5

row2: (5+10)/2=7.5

row3: (7.5+12)/2=9.75

thank you in advance!

Solution

EDIT: There is actually the method ewm implemented in pandas that can do this calculation

df['res'] = df['count'].ewm(alpha=0.5, adjust=False, min_periods=2).mean()

Original answer: Here is a way. as everything can be develop with coefficient being power of 2.

# first create a series with power of 2
coef = pd.Series(2**np.arange(len(df)), df.index).clip(lower=2)

df['res'] = (coef.div(2)*df['count']).cumsum()/coef

print(df)
   index  count   res
0      0      4  2.00
1      1      6  5.00
2      2     10  7.50
3      3     12  9.75

You can mask the first value with df.loc[0, 'res'] = np.nan if needed