df.rolling(7, center=True).mean()

The rolling mean can be centered by setting the center argument to True.

Example plot

Try the following (tested with pandas==0.23.3):

series.rolling('7D', min_periods=1, closed='left').sum().shift(-84, freq='h')

This will center your rolling sum in the 7-day window (by shifting -3.5 days), and will allow you to use a 'datetimelike' string for defining the window size. Note that shift() only takes an integer, thus defining with hours.

This will produce your desired output:

series.rolling('7D', min_periods=1, closed='left').sum().shift(-84, freq='h')['2014-01-01':].head(10)

2014-01-01 12:00:00    4.0
2014-01-02 12:00:00    5.0
2014-01-03 12:00:00    6.0
2014-01-04 12:00:00    7.0
2014-01-05 12:00:00    7.0
2014-01-06 12:00:00    7.0
2014-01-07 12:00:00    7.0
2014-01-08 12:00:00    7.0
2014-01-09 12:00:00    7.0
2014-01-10 12:00:00    7.0
Freq: D, dtype: float64

Note that the rolling sum is assigned to the center of the 7-day windows (using midnight to midnight timestamps), so the centered timestamp includes '12:00:00'.

Another option (as you show at the end of your question) is to resample the data to make sure it has even Datetime frequency, then use an integer for window size (window = 7) and center=True. However, you state that other parts of your code benefit from defining window with a 'datetimelike' string, so perhaps this option is not ideal.

See, it is supposed to work. Since, I don't have your dataset how it looks — I created myself

ser = pd.Series(np.random.randint(10,1000, 19), index=range(2001, 2020))

# Should look like this after your Group by
2001    578
2002    388
2003    803
2004    413
2005    125
2006    331
2007    179
2008    180
2009    331
2010    875
2011    422
2012    699
2013    256
2014    918
2015    566
2016    754
2017    521
2018    200
2019     16
dtype: int32

Now, doing the rolling:

ser.plot()
plt.ylim([0, df.max()])
ser.rolling(5, center=False).mean().plot()
plt.xticks(range(2000, 2020, 5));

The result is:

Now I think

You should get your data in the simple form like shown just above and store in the variable first — instead of stacking up all operations all together making one long line.

Then try the same. It should work.

I think you can use shift:

a = df.rolling(window=3).mean().shift(-2)
print (a)
           A
0   3.666667
1   5.666667
2  11.333333
3  18.333333
4        NaN
5        NaN