Simplifying Function with Recursive CTE and/or Window Function

I’m trying to come up with a Recursive CTE and/or Window Function to create a function.

After days, I’ve boiled the function down to (pseudocode) where I have N and B, and need to generate E:

En = Bn * (1 – SUM(E1, E2, … En-1))

Examples:

╔═══╦═════════════╦═════════════╗ ║ N ║ B           ║ E           ║ ╠═══╬═════════════╬═════════════╣ ║ 0 ║ 0.142857143 ║ 0.142857143 ║ ║ 1 ║ 0.285714286 ║ 0.244897959 ║ ║ 2 ║ 0.285714286 ║ 0.174927114 ║ ║ 3 ║ 0.285714286 ║ 0.124947938 ║ ║ 4 ║ 0.285714286 ║ 0.089248527 ║ ║ 5 ║ 0.4         ║ 0.089248527 ║ ║ 6 ║ 0.666666667 ║ 0.089248527 ║ ║ 7 ║ 1           ║ 0.044624264 ║ ╚═══╩═════════════╩═════════════╝ 

E0 = 0.143 * (1 – 0) = 0.143
E1 = 0.286 * (1 – 0.143) = 0.245
E2 = 0.286 * (1 – (0.143 + 0.245)) = 0.175
E3 = 0.286 * (1 – (0.143 + 0.245 + 0.175)) = 0.125
E4 = 0.286 * (1 – (0.143 + 0.245 + 0.175 + 0.125)) = 0.089
E5 = 0.400 * (1 – (0.143 + 0.245 + 0.175 + 0.125 + 0.089)) = 0.089
E6 = 0.667 * (1 – (0.143 + 0.245 + 0.175 + 0.125 + 0.089 + 0.089)) = 0.089
E7 = 1.000 * (1 – (0.143 + 0.245 + 0.175 + 0.125 + 0.089 + 0.089 + 0.089)) = 0.044

If the table above was in Excel, C2 = B2 * (1 - 0) (base) and C3 = B3 * (1 - SUM(C$ 2:C2)) (recursive)

What I’ve tried:

Windowed Functions

Tried SUM(...) OVER(ORDER BY [N] ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING), but can’t reference the column recursively.

Recursive CTE

Tried several iterations of:

WITH B AS ([Num], [Best], [Effective Rate]) AS (     SELECT *         , [Best]     FROM A     WHERE [Num] = 0     UNION ALL     SELECT A.*         , (1 - [Effective Rate]) * A.[Best]     FROM B     JOIN A ON A.[Num] = B.[Num] + 1  ) 

and some with an extra column in the CTE, but it only covers 1 previous row and results after 2nd row are wrong.

Recursive CTE with Windowed Function

From all that I’ve tried, it seems that the recursive segment of the CTE is calculated independently of the other results, and SUM(...) OVER(...) only works on the current row. (With regard to the above table, all values of E would be 0.142857143).

I assume this is because the UNION ALL happens all at once, and not incrementally.

Alternative Solutions

What I would really like to happen is to simplify the above equation, and/or transform it into an iterative function.

Bonus: If anyone cares to know the source of this information, it’s used to calculate MACRS depreciation for tax purposes.