I'm not sure what RDBMS you are using and I am aware this is the ANSI sql forum. However in DB2 you can use OLAP functions, but I don't know if these are ANSI compliant.
Anyway in DB2 the following should satisfy your query.
select date,
count,
sum(count) over (order by date) as rolling_sum
from table
order by date.
date count rolling_sum
01 1 1
02 2 3
03 7 10
04 10 20
Hope this helps, it works fine under DB2 as I said.