The value of ∑k=0n 4n+1Ck+4n+1C2n−k is
24n+4n+1C2n
24n+1
24n
24n+1+4n+1Cn
∑k=0n 4n+1Ck+4n+1C2n−k=∑k=02n 4n+1Ck+4n+1C2n=24n+4n+1C2n