After noticing that the first three terms of OEIS A016269 coincide with those of OEIS A030053 and the fourth differs only by $1$, and working with successive approximations, I obtained the following identity, valid for $0 \le n \le 26$, where the binomial coefficients $\binom{m}{k}$ evaluate to $0$ for $k \lt 0$:
$$2^{n+1}(2^{n+2}+1)-3^{n+2} ={2n+7 \choose n}-{2n+6 \choose n-3}+{2n+7 \choose n-5}+{2n+7 \choose n-12}-{2n+6 \choose n-15}+{2n+7 \choose n-17}+{2n+7 \choose n-24}$$
The series is somewhat "irregular", but it could be expanded further.
Forcing $m=2n+7$ always in $\binom{m}{k}$ we get a more "regular" sequence but with more terms:
$$2^{n+1}(2^{n+2}+1)-3^{n+2} ={2n+7 \choose n}-{2n+7 \choose n-3}+{2n+7 \choose n-4}+{2n+7 \choose n-6}-{2n+7 \choose n-7}+{2n+7 \choose n-8}+{2n+7 \choose n-9}$$
valid for $0 \le n \le 9$. This one implies that for $n \neq k_i$ and $2n+7$ prime, all the $\binom{2n+7}{n-k_i}$ terms are divisible by $2n+7$ and therefore $2n+7$ divides the LHS.
What is special about $2^{n+1}(2^{n+2}+1)-3^{n+2}$ that it can be expanded in this way? And what is special about $2n+7$? Also is there some series expansion (maybe based on the gamma function?) of general application that I don't know?