Calculate rolling sum over 7 consecutive day period on PostgreSQL
By far the cleanest solution is to use window function sum with rows between:

with days as (
        SELECT date_trunc('day', d)::date as day
        FROM generate_series(CURRENT_DATE-31, CURRENT_DATE-1, '1 day'::interval) d ),
    counts as (
        select 
            days.day,
            sum((random()*5)::integer) num
        FROM days
        -- left join other tables here to get counts, I'm using random
        group by days.day
    )
select
    day,
    num,
    sum(num) over (order by day ROWS BETWEEN 6 PRECEDING AND CURRENT ROW)
from counts
order by day;


The important part is to generate the timeframe in days CTE and join onto it in order not to miss any days for which there is no data.

Example

For example, if I create some test data with 20 records in last 14 days:

SELECT (current_date - ((random()*14)::integer::text || 'days')::interval)::date as day, (random()*7)::integer as num
into test_data from generate_series(1, 20);;


And also add a value before that:

insert into test_data values ((current_date - '25 days'::interval), 5);


Then use the query above:

with days as (
        SELECT date_trunc('day', d)::date as day
        FROM generate_series(CURRENT_DATE-31, CURRENT_DATE-1, '1 day'::interval) d ),
    counts as (
        select 
            days.day,
            sum(t.num) num
        FROM days
        left join test_data t on t.day = days.day
        group by days.day
    )
select
    day,
    num,
    sum(num) over (order by day rows between 6 preceding and current row)
from counts
order by day;


And get the results for whole month:

    day     | num | sum 
------------+-----+-----
 2016-01-31 |     |    
 2016-02-01 |     |    
 2016-02-02 |     |    
 2016-02-03 |     |    
 2016-02-04 |     |    
 2016-02-05 |     |    
 2016-02-06 |   5 |   5
 2016-02-07 |     |   5
 2016-02-08 |     |   5
 2016-02-09 |     |   5
 2016-02-10 |     |   5
 2016-02-11 |     |   5
 2016-02-12 |     |   5
 2016-02-13 |     |    
 2016-02-14 |     |    
 2016-02-15 |     |    
 2016-02-16 |     |    
 2016-02-17 |     |    
 2016-02-18 |   2 |   2
 2016-02-19 |   5 |   7
 2016-02-20 |     |   7
 2016-02-21 |   4 |  11
 2016-02-22 |  15 |  26
 2016-02-23 |   1 |  27
 2016-02-24 |   1 |  28
 2016-02-25 |   2 |  28
 2016-02-26 |   4 |  27
 2016-02-27 |   9 |  36
 2016-02-28 |   5 |  37
 2016-02-29 |  11 |  33
 2016-03-01 |   5 |  37
(31 rows)