algebra problem

A question for mathematical experts on the list related to my efficiency
equations:

I'm simplifying sigmas out of the equations I have, which I can do for
terms containing only multiples or powers of the bound variable, but I run
into trouble with factorials.
How would you translate "summation from m=1 to n (m!)" into an algebraic
expression? The series as n increases is 1, 3, 9, 33, 153, 873, 5913 . . .
and it can be written as
1 n = 1
1(1 + 2) n = 2
1(1 + 2(1 + 3)) n = 3
1(1 + 2(1 + 3(1 + 4)) n = 4
and so on.

jason