(*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][1] 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,

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

  [1]: https://www.researchgate.net/publication/257642843_Sporadic_sequences_modular_forms_and_new_series_for_1p