Who wrote the formula for gini importance/sklearn's feature importance score? 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
There is a paper that covers "The origins of the Gini index". Gini index was detailed by Leo Breiman et al. in "Classification and regression trees" book in 1984. Leo Breiman also wrote a seminal paper on Random Forests in 2001 which includes the notion of feature importance.