I'm wondering whether the following polynomial of a single indeterminate has been studied: take the (partial) Bell polynomial $B_{n,k}$, which is a polynomial in indeterminates $x_1$, $x_2$, …, $x_{n-k+1}$, and replace each indeterminate $x_i$ by the falling factorial $(x)_i=x(x-1) \dots (x-i+1)$. Call this polynomial $N(n,k)$.

I conjecture the following matrix identity. Assemble the polynomials into an infinite lower-triangular matrix $N$. Let $s$ be the lower-triangular matrix of Stirling numbers of the first kind. Let D be the diagonal matrix with entries $x$, $x^2$, $x^3$, ….

Conjecture: $Ns=sD$