# Context Sensitive Grammar for the language $\{ a^{2n} b^{2n+1} c^{3n} d^{n+3} \mid n \ge 1\}$

I have been trying to find a context sensitive grammar for the language $$\{ a^{2n} b^{2n+1} c^{3n} d^{n+3} \mid n \ge 1\}$$ for some time but I cannot get it done. Any ideas ?

In the following grammar the first block of productions ensures that the right amount of $$a,b,c$$, and $$d$$ is generated, in some order. The terminal $$a$$ is represented by the nonterminal $$A$$, $$b$$ by $$B$$, etc. They also ensure that $$X$$, which represents a $$a$$, is at the end of the sentential form.
The next block ensures that $$A,B,C,D$$ can be reordered.
The last block ensures that terminals are generated from right to left in the correct order (first $$d$$, then $$c$$, then $$b$$, and finally $$a$$).
\begin{align*} S &\to ABBCCCDS'X \\ S' &\to AABBCCCDS' \mid BDDD\\ \\ BA & \to AB \\ CA & \to AC \\ CB & \to BC \\ DA & \to AD \\ DB & \to BD \\ DC & \to CD \\ \\ DX &\to Xd \\ X &\to Y \\ CY &\to Yc \\ Y &\to W \\ BW &\to Wb \\ W &\to Z \\ AZ &\to Za \\ Z & \to a \end{align*}
• Can you provide a sentence that is in the language but is not generated by the grammar, or a derivation of a sentence that is not in the language? Also there was a spurious $S$ in right part the second production. I removed it. – Steven Apr 15 '20 at 13:22