How to add a condition to a double summation

Question

I want to write the following equation in Mathematica, but I don't know how to do that

Answer 1

One of the ways is as follows.

\[Lambda][1]* Sum[(-1)^(n - 1)*Sum[\[Lambda][1]^i*\[Lambda][2]^j*
 KroneckerDelta[i + j - n + 1], {i, 0, n - 1}, {j, 0, n - 1}]*
 t^n/n!, {n, 1, Infinity}]

$-\frac{\lambda (1) e^{\lambda (1) (-t)-\lambda (2) t} \left(e^{\lambda (2) t}-e^{\lambda (1) t}\right)}{\lambda (1)-\lambda (2)}$