Weighted hierarchical cumulative sum

I don't know if I'm using the correct terminology, but I'm trying to find the cumulative sum of each node in a tree structure, but the children node only contribute a percentage of its cumulative sum to its parent node's cumulative sum.

Emission(E1) = Emission(E1)
Emission(E2) = Emission(E2) + percent_Contribution(E1) * Emission(E1)
Emissions(E3) = Emission(E3) + percent_Contribution(E2) * Emission(E2)

This a sample of the result I want to achieve:

Note that this is a sample of the result I'm trying to achieve, as my actual table can have many more levels of hierarchy.

I'm unfamiliar with SQL, but my research tells me that I should right a recursive CTE, but I haven't been able to get it to work with the weighted value. If it's impossible to do with pure SQL, I would also appreciated being point toward the right direction.

Edited Table in case of parent having multiple child records:

Solution with new sorting

WITH RECURSIVE
cte AS (
  SELECT entityId, parentId, emission, percentContribution,  CAST(emission as DECIMAL(3,1)) as cumulativeSum
  FROM Emissions WHERE entityId='E1'  
  UNION ALL
  SELECT e.entityId, e.parentId, e.emission, e.percentContribution,  e.emission + (c.cumulativeSum * c.percentContribution /100) 
  FROM cte c
  INNER JOIN Emissions e ON e.entityId=c.parentId
  WHERE c.parentId is not null
)
SELECT * FROM cte ORDER BY entityId;

fiddle