Number of way to render an amount of money
Java 1.5 was released in 2004, use generics for collections.
Consider making  array (terrible variable name BTW) and memo fields of object that does the computation instead of passing them as arguments every time
Why is amountOfMoneyToRender a double instead of int or long?
String.valueOf is unnecessary for i + "some literal"
Instead of using containsKey + get just do get and check for null.
It might be good idea to change loop into for(int j = i; j &lt; array.length; j++), it will make easier to spot bug in your code (with amountOfMoneyToRender=10, the result should be 11, not 100)