Another Recursion Puzzle

[Leroy Quet]

Let a(0) = 0;
Let a(1) = 1;

Let, for m >= 2,

a(m) = a(m-1) *m*(m+1) - a(m-2) *(m-1)*m - m! *(m-1)! + 1

Give a closed form (in terms of a partial sum) for the sequence of a(m)'s.

(This sequence is not in the E.I.S. yet.)

(I'll give my answer in a couple of days if no one else gets it sooner, 
of course.)

Thanks,
Leroy Quet