Always bring your data as plain text, not in an image, so we all can copy/paste.
I took a total of 15 minutes to re-type your input data and re-format it.
But, now I have it:
What you need is the running sum of: 1 if it's a week day, 0 if it's a week-end-day, per month. That's plain vanilla OLAP , window function, functionality. Welcome to SQL-99 ...
Here goes.
The input:
WITH
-- your input as in-line table, typed manually ...
input(index,price,pricedate,counta_in) AS (
SELECT 'A',NULL::INT,DATE '2020-05-01', 1
UNION ALL SELECT 'A', 5,DATE '2020-05-04', 2
UNION ALL SELECT 'A', 4,DATE '2020-05-05', 3
UNION ALL SELECT 'A', 6,DATE '2020-05-06', 4
UNION ALL SELECT 'A', 4,DATE '2020-05-07', 5
UNION ALL SELECT 'A', 4,DATE '2020-05-11', 6
UNION ALL SELECT 'A', 6,DATE '2020-05-12', 7
UNION ALL SELECT 'A', 7,DATE '2020-05-13', 8
UNION ALL SELECT 'A', 8,DATE '2020-05-14', 9
UNION ALL SELECT 'A', 8,DATE '2020-05-15',10
UNION ALL SELECT 'A', 8,DATE '2020-05-18',11
UNION ALL SELECT 'A', 4,DATE '2020-05-19',12
UNION ALL SELECT 'A', 5,DATE '2020-05-20',13
UNION ALL SELECT 'A', 3,DATE '2020-05-21',14
UNION ALL SELECT 'A', 5,DATE '2020-05-22',15
UNION ALL SELECT 'A', 4,DATE '2020-05-26',16
UNION ALL SELECT 'A', 5,DATE '2020-05-27',17
UNION ALL SELECT 'A', 3,DATE '2020-05-28',18
UNION ALL SELECT 'A', 5,DATE '2020-05-29',19
UNION ALL SELECT 'A', 4,DATE '2020-06-01', 1
UNION ALL SELECT 'A', 6,DATE '2020-06-02', 2
UNION ALL SELECT 'A', 4,DATE '2020-06-03', 3
UNION ALL SELECT 'A', 4,DATE '2020-06-04', 4
UNION ALL SELECT 'A', 5,DATE '2020-06-05', 5
UNION ALL SELECT 'A', 4,DATE '2020-06-08', 6
UNION ALL SELECT 'A', 6,DATE '2020-06-09', 7
)
The actual select, selecting from the input above:
-- acutal query starts here. counta_out is my OLAP expression.
SELECT
*
, SUM(CASE WHEN DAYOFWEEK_ISO(pricedate) < 6 THEN 1 ELSE 0 END)
OVER(PARTITION BY MONTH(pricedate) ORDER BY pricedate)
AS counta_out
FROM input;
And the result, including control column and result column:
index | price | pricedate | counta_in | counta_out
------+-------+------------+-----------+------------
A | | 2020-05-01 | 1 | 1
A | 5 | 2020-05-04 | 2 | 2
A | 4 | 2020-05-05 | 3 | 3
A | 6 | 2020-05-06 | 4 | 4
A | 4 | 2020-05-07 | 5 | 5
A | 4 | 2020-05-11 | 6 | 6
A | 6 | 2020-05-12 | 7 | 7
A | 7 | 2020-05-13 | 8 | 8
A | 8 | 2020-05-14 | 9 | 9
A | 8 | 2020-05-15 | 10 | 10
A | 8 | 2020-05-18 | 11 | 11
A | 4 | 2020-05-19 | 12 | 12
A | 5 | 2020-05-20 | 13 | 13
A | 3 | 2020-05-21 | 14 | 14
A | 5 | 2020-05-22 | 15 | 15
A | 4 | 2020-05-26 | 16 | 16
A | 5 | 2020-05-27 | 17 | 17
A | 3 | 2020-05-28 | 18 | 18
A | 5 | 2020-05-29 | 19 | 19
A | 4 | 2020-06-01 | 1 | 1
A | 6 | 2020-06-02 | 2 | 2
A | 4 | 2020-06-03 | 3 | 3
A | 4 | 2020-06-04 | 4 | 4
A | 5 | 2020-06-05 | 5 | 5
A | 4 | 2020-06-08 | 6 | 6
A | 6 | 2020-06-09 | 7 | 7