$$
f_n = \sum_{i=0}^n (-1)^i {n \choose i} g_i \Leftrightarrow g_n = \sum_{i=0}^n (-1)^i {n \choose i} f_i
$$

$$
f_n = \sum_{i=0}^n {n \choose i} g_i \Leftrightarrow g_n = \sum_{i=0}^n (-1)^{n-i} {n \choose i} f_i
$$

$$
f_k=\sum_{i=k}^n {i\choose k} g_i\Leftrightarrow g_k=\sum_{i=k}^n (-1)^{i-k} {i\choose k} f_i
$$