Compute the number of ways a given amount (cents) can be changed

I don't see any correctness issues, but it could be more idiomatic.

 if coin_index in memoized_sol: if rem_cents in memoized_sol[coin_index]: return memoized_sol[coin_index][rem_cents] else: memoized_sol[coin_index] = {} 

Check out first defaultdict, and then functools.lru_cache for ways to simplify the memoisation - although note that lru_cache has a subtlety in that it doesn't like list as an argument type.

 if rem_cents == 0: return 1 if coin_index >= len(coin_vals): return 0 

IMO these should have gone before the memoisation code, because they are non-memoised special cases. But I freely admit that this is mainly a matter of opinion. The only real difference effected would be to avoid initialising memoized_sol[len(coin_vals)].

 i = 0 count = 0 while i*coin_val <= rem_cents: count = count + compute_coin_change_count\ ( rem_cents - i*coin_val, coin_vals, coin_index+1, memoized_sol ) i = i + 1 

Using sum and range this could be simplified to

count = sum(compute_coin_change_count( surplus, coin_vals, coin_index + 1, memoized_sol) for surplus in range(rem_cents, -1, -coin_vals[coin_index])) 

(I'm not quite sure what the most Pythonesque indentation would be - I wouldn't normally follow PEP8 on maximum line lengths).

I would also be tempted to use sublists and eliminate the need for coin_index, but that depends on how you're handling memoisation.

Putting it all together, I get

from functools import lru_cache def get_coin_change_count(cents, coin_vals): @lru_cache(None) def inner(remaining, idx): if remaining == 0: return 1 if idx == len(coin_vals): return 0 return sum(inner(surplus, idx + 1) for surplus in range(remaining, -1, -coin_vals[idx])) return inner(cents, 0) print(get_coin_change_count(37, [10, 1, 5, 25])) 

and here the indentation does pass PEP8, in part because of shorter names. Note that using an inner function allows memoisation with lru_cache and allows the short name inner because its scope is restricted enough that this is sufficiently expressive.