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We present combinatorial and analytical results concerning a Sheffer sequencewith an exponential generating function of the form $G(s,z)=e^{czs+\alphaz^{2}+\beta z^{4}}$, where $\alpha, \beta, c \in \mathbb{R}$ with $\beta<0$ and$c\neq 0$. We demonstrate that the zeros of all polynomials in such a Sheffersequence are either real, or purely imaginary. Additionally, using theproperties of Riordan matrices we show that our Sheffer sequence satisfies athree-term recurrence relation of order 4, and we also exhibit a connectionbetween the coefficients of these Sheffer polynomials and the number of nodeswith a a given label in certain marked generating trees.

The zero locus and some combinatorial properties of certain exponential Sheffer sequences