Per S. Cooper's request, a table relating level $1$ with level $9$ can be provided. The general form apparently is,

$$\frac{1}{\pi}=\frac{12}{(C)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k} \tbinom{3k}{k}\tbinom{6k}{3k} \frac{3A\,k+B}{(-C^3)^k}$$ and, $$\frac{1}{\pi}=\frac{12}{(C-4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j} \frac{A\,k+B+N\color{blue}\alpha/3}{(-C+4\color{blue}\alpha)^k}$$

for general real $\color{blue}\alpha$ and where,

$$\begin{array}{|c|c|c|c|c|} \hline d&A&B&C&N\\ \hline 11&154/9&5&32&4/3\\ 19&114&25&96&4\\ 43 &5418 &789 &960 &24\\ 67 &87234 &10177 &5280 &76\\ 163 &181713378 &13591409 &640320 &1448\\ \hline \end{array}$$

The variables $A,B,C$ are known to have closed-form expressions in terms of $d$. Presumably $N$ should have as well.

P.S.1: Note also the equivalent forms,

$$s_k(\color{blue}\alpha)=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j}=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{j}\tbinom{k-j}{j}\tbinom{k-2j}{j}$$

where the latter form is used in H. Chan and S. Cooper's paper "Rational analogues of Ramanujan's series for 1/π". The case $\alpha=-3$,

$$s_k(-3) = 1, -3, 9, -21, 9, 297, -2421$$

is one of the six sporadic sequences studied by Zagier and Cooper.

P.S.2: A similar table can be created relating level $2$ with level $8$.

(Per S. Cooper's request.)

I. Table relating level $1$ with level $9$.

The general form apparently is,

$$\frac{1}{\pi}=\frac{12}{(C)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k} \tbinom{3k}{k}\tbinom{6k}{3k} \frac{\color{red}3A\,k+B}{(-C^3)^k}$$ and, $$\frac{1}{\pi}=\frac{12}{(C-4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j} \frac{A\,k+B+N\color{blue}\alpha/3}{(-C+4\color{blue}\alpha)^k}$$

for general real $\color{blue}\alpha$ and where,

$$\begin{array}{|c|c|c|c|c|} \hline d&A&B&C&N\\ \hline 11&154/9&5&32&4/3\\ 19&114&25&96&4\\ 43 &5418 &789 &960 &24\\ 67 &87234 &10177 &5280 &76\\ 163 &181713378 &13591409 &640320 &1448\\ \hline \end{array}$$

The variables $A,B,C$ are known to have closed-form expressions in terms of $d$. Presumably $N$ should have as well.

P.S. Note also the equivalent forms,

$$s_k(\color{blue}\alpha)=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j}=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{j}\tbinom{k-j}{j}\tbinom{k-2j}{j}$$

where the latter form is used in H. Chan and S. Cooper's paper "Rational analogues of Ramanujan's series for 1/π". The case $\alpha=-3$,

$$s_k(-3) = 1, -3, 9, -21, 9, 297, -2421$$

is one of the six sporadic sequences studied by Zagier and Cooper.

II. Table relating level $2$ with level $8$.

The general form apparently is,

$$\frac{1}{\pi}=\frac{192\sqrt{2}}{(C)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k} \tbinom{2k}{k}\tbinom{4k}{2k} \frac{\color{red}2A\,k+B}{(C^2)^k}$$ and, $$\frac{1}{\pi}=\frac{192\sqrt{2}}{(C+4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-2j} \tbinom{k}{2j}\tbinom{2j}{j}\tbinom{2j}{j} \frac{A\,k+B-M\color{blue}\alpha/4}{(C+4\color{blue}\alpha)^k}$$

for general real $\color{blue}\alpha$ and where,

$$\begin{array}{|c|c|c|c|c|} \hline d&A&B&C&M\\ \hline 6&\sqrt2&\sqrt2/4&(4\sqrt3)^2&\sqrt2/12\\ 10&10&2&12^2&1/3\\ 18 &70\sqrt6 &21\sqrt6/2 &28^2 &\sqrt6/2\\ 22 &385\sqrt2 &209\sqrt2/4 &(12\sqrt{11})^2 &17\sqrt2/12\\ 58 &870870&72798&396^2&37\\ \hline \end{array}$$