I've been looking for a paper where the Gini importance was first proposed, but I am not sure if this is actually how it came to be.
Here's the formula I am familiar with and am looking to find in a paper:
$$\frac{N_s}{N_t} * \left(i - \frac{{N_s}_r}{N_s} i_r - \frac{{N_s}_l}{N_s} i_l \right)$$
where
$N_s$ = number of samples at a particular node
$N_t$ = number of total samples
$i$ = Impurity
$*_r$ = measure of the right child node
$*_l$ = measure of the left child node
If I understand properly, this is the formula sklearn's random forests also use in model.feature_importances_
Tags (since I don't have the rep to create new ones yet): Feature Importance Score, Gini Importance