The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,

$$(g(x)D_x)^n.$$

The formula section of A139605 contains the matrix formula. Multiply the $n$-th diagonal (with $n=0$ the main diagonal) of the lower triangular Pascal matrix A007318 by $g_n = D_x^n g(x)$ to obtain the matrix $VP$ with $VP_{n,k} = \binom{n}{k}g_{n-k} $. Then $$(g(x)D)^n = (1, 0, 0,..) [VP \dot \; S]^n (1, D, D^2, ..)^T,$$ where S is the shift matrix A129185, representing differentiation in the divided powers basis $x^n/n!$.

Example:

$$(g(x)D_x)^3$$

$$= (1, 0, 0, 0) [VP \dot \; S]^3 (1, D, D^2, D^3)^T$$

$$= \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & g_0 & 0 & 0 \\ 0 & g_1 & g_0 & 0\\ 0 & g_2 & 2g_1 & g_0 \\ 0 & g_3 & 3g_2 & 3g_1 \end{pmatrix}^3 \begin{pmatrix} 1 \\ D \\ D^2 \\ D^3 \end{pmatrix} $$

$$ = [g_0g_1^2 + g_0^2 g_2] D + 3 g_0^2g_1 D^2 + g_0^3D^3 $$

And, the pdf Mathemagical Forests gives a diagrammatic method for creating forests of trees through "natural growth" that represent the partition polynomials.

I have either lost a proof of the validity of this formula or got sidetracked before I developed one.

**Question**:

Can someone prove this conjecture?

**Some background:**

The refined Eulerian numbers (RENs) of A145271 are related analytically to the compositional inversion of functions and formal generating series and to flow fields generated by tangent vectors. The $n$-th row of RENs are the numerical coefficients of the expansion of $(g(x)\frac{d}{dx})^ng(x)$ in terms of the monomials in the derivatives of $g(x)$, i.e.,

$$g_k=\frac{d^k}{dx^k}g(x).$$

For example,

$$(g(x)\frac{d}{dx})^3g(x) = 1 g_0^1 g_1^3 + 4 g_0^2 g_1^1 g_2^1 + 1 g_0^3 g_3^1.$$

With $(\omega,x) = (f(x),f^{(-1)}(\omega))$ and $g(x) = 1/f^{'}(x)$,

$$\exp[t g(x)d/dx]x = \exp[td/d\omega]f^{(-1)}(\omega) = f^{(-1)}(t+\omega)=f^{(-1)}(t+f(x)).$$

Evaluated at the origin of $x$, this gives the compositional inverse

$$\exp[tg(x)d/dx] x |_{x=0}=f^{(-1)}(t).$$

See also

1) MO-Q Guises of the refined Eulerian numbers generated by tangent vectors

2) MO-Q Important formulas in combinatorics

3) MO-Q Why is there a connection between enumerative geometry and nonlinear waves?