Asking for a closed form solution may be too much. You can calculate values:
a[1] = 1;
a[n_] := a[n] = (n!/(\[Lambda] - \[Lambda]^n))*
Sum[(a[n - k]*\[Lambda]^(n - k)*(-1)^k)/(k!*(n - 2 k)!), {k, 1,
Floor[n/2]}];
TableForm@Table[Simplify@{n, a[n]}, {n, 1, 10}]
