The factorials are:

1,

2 x 1 = 2,

3 x 2 x 1 = 6,

4 x 3 x 2 x 1 = 24,

5 x 4 x 3 x 2 x 1 = 120,

6 x 5 x 4 x 3 x 2 x 1 = 720....

The 3 digit numbers less than 500 which are sums of consecutive
factorials are:

144, 150, 152 and 153.

The only one of these numbers which satisfies the condition that it is equal to the sum of the cubes of its digits is 153 because 153 = 1 + 125 + 27 so this must be the forgotten number. The information about triangular numbers was not needed but it is easily checked that 153 is a triangular number.

An excellent solution was recieved from Ben from Methwold High School.