In the paper, by the Faà di Bruno formula, the authors establish two explicit formulas for the Motzkin numbers, the generalized Motzkin numbers, and the restricted hexagonal numbers.

For sufficiently differentiable univariate functions f, g and their composite h = g(f) we prove that dg(f(x)) df(x)n = n ∑

We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy.

We derive a formula for the number of non-negative integer solutions equation $a_1x_1 + a_2x_2 \cdots a_mx_m = n$ in terms partial Bell polynomials via Faa di Bruno’s formula.

Derivatives of a solution of an ODE Cauchy problem can be computed inductively using the Faà di Bruno formula. In this paper, we exhibit a noninductive formula for these derivatives. At the heart of this formula is a combinatorial problem, which is solved in this paper. We also give a more tractable form of the Magnus expansion for the solution of a homogeneous linear ODE.

We present a self-contained account of recent results on moment identities for Poisson stochastic integrals with random integrands, based on the use of functional transforms on the Poisson space. This presentation relies on elementary combinatorics based on the Faà di Bruno formula, partitions and polynomials, which are used together with multiple stochastic integrals, finite difference operato...

Explicit and recursive formulas for Bernoulli and Euler numbers are derived from the Faá di Bruno formula for the higher derivatives of a composite function. Along the way we prove a result about composite generating functions which can be systematically used to derive such identities.

In this paper we use Faa di Bruno's formula to associate Bell polynomial values differential equations of the form $y^{\prime}=f(y)$. That is, partial polynomials represent solution such an equation and compute special polynomials.

In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which incl...