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Let $(a_n)_{n \geq 0}$ be a sequence of integers such that its generatingseries satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for somepolynomial $h(t)$. For any $r \geq 1$ we study the coefficient sequence of thenumerator polynomial $h_0(a^{<r >}) +...+ h_{\lambda'}(a^{<r >}) t^{\lambda'}$of the $r$\textsuperscript{th} Veronese series $a^{<r >}(t) = \sum_{n \geq 0}a_{nr} t^n$. Under mild hypothesis we show that the vector of successivedifferences of this sequence up to the $\lfloor \frac{d}{2}\rfloor$\textsuperscript{th} entry is the $f$-vector of a simplicial complexfor large $r$. In particular, the sequence satisfies the consequences of theunimodality part of the $g$-conjecture. We give applications of the main resultto Hilbert series of Veronese algebras of standard graded algebras and the$f$-vectors of edgewise subdivisions of simplicial complexes.

Enumerative $g$-theorems for the Veronese construction for formal power series and graded algebras