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Your conjecture is true. First, observe that \begin{align*} K_{w+1}(z)&=(1+z)\,K_w(z) & \mbox{ if } w \mbox{ is odd, }\\ K_{w+1}(z)&=(1+z)\,K_w(z) +\frac{1}{2}{w\choose \frac{w}{2}}z^{w/2}(1-z) & …

Here is a way. Start from the exact form of your integral expression (i.e. supply the "Dirichlet factor" $\frac{\Gamma(\alpha_1+\ldots+\alpha_n)}{\Gamma(\alpha_1)\cdots\Gamma(\alpha_n)})$. (1) Writing …

These assertions can be proved using (formal) generating functions. Using that for $j\geq 0, k\geq 1$ \begin{align*} \sum_{n\geq 0} {n-j+kj \choose kj} t^n &=\frac{t^j}{(1-t)^{kj+1} }\;\;\mbox{ …

A generating function proof. As $\arcsin(z)=\sum_{k\geq 0} \frac{1}{2k+1} {2k \choose k} \frac{z^{2k+1}}{4^k}$ and $\frac{1}{\sqrt{1-z^2}}=\sum_{k\geq 0}{2k \choose k}\frac{z^{2k}}{4^k}$ we have that …

This question was completely settled by J.A. Fill here: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.641

As in Timothy Budd's answer let $w=w(q)$ denote the (formal) solution of $q=\frac{w}{f(w)}$. Let $p$ be another variable and consider the sum \begin{align*} S(p,q):=\sum_{n,m>0} \sum_{j>0} j[z^{n+j}]\ …

The following inequality holds: $$\mathbb{P}(C_k(m)\geq 1)\geq \mathbb{P}(\mathrm{Bin}(m, \lVert A\rVert_k)\geq k)$$ where here and in the sequel $\mathrm{Bin}(n,p)$ denotes a binomially distributed r …

One can also use the binomial transform. (If $A(z)=\sum_{i\geq 0} a_i z^i$ is a (formal) power series, the (formal) power series $B(z):=\frac{1}{1-z} A(\frac{z}{1-z})$ has coefficients $[z^n] B(z)=\su …

There a well known generating function methods (the ''symbolic method'' and ''Poissonization'') which can be used to deal with this kind of question. However, I am unable to point to a reference for …

By @Max Alexeyev's solution above $N_{2k-1}^{\sigma}=tr(M_{k}(P_{\sigma}M_{k}P_{\sigma}^{-1}))$. The eigenvalues and eigenvectors of $M_k$ are given here: Result attribution for eigenvalues of a matri …

$\lambda(f):=\kappa_f-1$ is called "the coefficient of coalescence of $f$" here: https://msp.org/pjm/1982/103-2/pjm-v103-n2-p03-p.pdf (note the typo on p.269, the correct definition appears on p.27 …

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.) A simple way is to use …

Here is a proof using (formal) generating functions. The Lah number $L(n,m+1)$ $$L(n,m+1)=\frac{n!}{(m+1)!}[x^n] \bigg(\frac{x}{1-x}\bigg)^{m+1}$$ counts the number of unordered partitions of the se …

(Not a solution, just a reformulation and a conjecture for $w=3$) (1) Remark: the question above may equivalently be stated as follows: Let $Z$ be the matrix of the cyclic shift (the companion matri …

Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\ma …