summations with unusual notation,as interpreted?

Question

I have seen here and in many places summations that do not follow a standard notation, I put some of them below: For example how to interpret these. Some bibliography with this type of notation? I can't find anything other than usual

Answer 1

The first does not appear to be a sum at all unless interpreted as on the right. The others also appear to be coded by people without a strong background in standard notation. These are just educated guesses about what the coders intended.

\begin{align*} \sum_{\large{r_1+r_2+\cdots+r_n = t,\space r_k \in\mathbb{N}}} \frac{t!}{r_1!r_2!\cdots r_n!} \\ \longrightarrow\space \sum_{\Large{t=r_1}}^{\Large{r_n,\space r_k\in\mathbb{N}}} \frac{t!}{r_1!r_2!\cdots r_n!} \end{align*}

\begin{align*} \sum_{\large{1<t<k\le n}} \ln\left(\ln\left(\sqrt[\huge{n}]{\frac{3n-2k}{3n+2k}}\right)\right) \\\longrightarrow\space \sum_{\large{k>t>1}}^{\Large{n}} \ln\left(\ln\left(\sqrt[\huge{n}]{\frac{3n-2k}{3n+2k}}\right)\right) \end{align*}

\begin{align*} \sum_{\large{k\ge0} } (tk)^{k-1}\frac{z^k}{k!} \quad\longrightarrow\quad \sum_{\large{k=0} }^{\Large{\infty}} (tk)^{k-1}\frac{z^k}{k!} \end{align*}

\begin{align*} \sum_{\large{d|n}} g(d) \quad\longrightarrow\quad \sum_{\large{\frac{n}{d}=1}}^{\Large{\infty}, \space \frac{n}{d}\in\mathbb{N}} \space g(d) \end{align*}