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Let $a(n)$ be A302117. Here $$ a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\ a(0) = 0. $$ Let $$ T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}. $$ Let $P_n(z)$ be the family of ...

Recall that a Pisot polynomial $P=x^n+ a_{n-1}x^{n-1}\ldots a_1$ has integer coefficients, a real root $x_1>1$ and all other roots $|x_i|<1$ for $1\leq i \leq n$. One key result is that $\{(...

Let $\mathbb{N}_+ = \{ 1, 2, \dots\} $. For a given sequence of elements $\{a_i \}_{1 \leq i \leq m} $in $ \mathbb{N}_+ $, we define \begin{equation} P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \...

It is known that Striling numbers of the second kind satisfy the relation $$ \sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n. $$ where $(x)_n$ is the falling factorials such that $$ (x)_n = x(x-1)(x-2)\...