<s>[INST] General method to find sum of binomial Specifically I want to ask the method to solve
$$\sum_{k=0}^n \binom{4n+b}{4k};\ b=[0,1,2,3]$$
And how to solve series of type
$$\sum_{k=0}^n \binom{an+b}{ak};\ a=[1,2,3,...],\ b=[0,1,2,...,a-1]$$
 [/INST] I am sure that you will enjoy the result !
If  $$S_n=\sum_{k=0}^n \binom{4n+b}{4k}$$ then a CAS gives
$$4S_n=2^{\frac{4 n+b}{2} } \left((-1)^b\, 2^{\frac{4 n+b}{2} } \cos (\pi  b)+\cos
   \left(\frac{  (4 n+b)\pi}{4} \right)+(-1)^b \cos \left(\frac{3(4
   n+b)\pi}{4}  \right)\right)-$$ $$4 \binom{4 n+b}{4 n+4} \,
   _5F_4\left(1,\frac{4-b}{4},\frac{5-b}{4},\frac{6-b}{4},\frac
   {7-b}{4};\frac{4n+5}{4},\frac{4n+6}{4},\frac{4n+7}{4},\frac{4n+8}{4};1\right)$$ where appears the generalized hypergeometric function.
For the first values of $b$, the sequences can be found in $OEIS$.
$$\left(
\begin{array}{cc}
 b & \text{OEIS sequence} \\
 0 & \text{A070775} \\
 1 & \text{A090407} \\
 2 & \text{A001025} \\
 3 & \text{A090408}
\end{array}
\right)$$
Be sure that I did not make this by myself.
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