How to calculate the sum of general type $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r } $$ ?

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a}$$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} (2 k - n + a) = 0 $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 2} = \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 3} = - \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$ . I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 4} $$ Can someone help me with this one?

QUESTION. How to calculate the sum of such general type? $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r }. $$

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a},$$ $$\sum_{k=0}^n {n\choose k} {n\choose k+a} \binom{2 k - n + a}1 = 0, $$ $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 2} = -\frac{(a+ n)(a-n)}{2 (2n-1)} {2n\choose n+a}, $$ $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 3} = \frac{(a+ n)(a-n)}{2 (2n-1)} {2n\choose n+a}. $$ I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 4} $$ Can someone help me with this one?

How to calculate the sum of general type $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r } $$ ?

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a}$$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} (2 k - n + a) = 0 $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 2} = \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 3} = - \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$ . I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 4} = ? $$ Can someone help me with this one?

How to calculate the sum of general type $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r } $$ ?

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a}$$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} (2 k - n + a) = 0 $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 2} = \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 3} = - \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$ . I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 4} $$ Can someone help me with this one?

How to calculate the sum of general type $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 l - n + a \choose r } $$ ?

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a}$$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} (2 l - n + a) = 0 $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 l - n + a \choose 2} = \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 l - n + a \choose 3} = - \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$ . I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 l - n + a \choose 4} = ? $$ Can someone help me with this one?

How to calculate the sum of general type $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r } $$ ?

Some particular examples $$\sum_{k=0}^n {n\choose k} {n\choose k+a} = {2n\choose n+a}$$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} (2 k - n + a) = 0 $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 2} = \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$, $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 3} = - \frac{(a+ n)(a-n)}{2 (1-2 n)} {2n\choose n+a} $$ . I am struggling to calculate at least $$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose 4} = ? $$ Can someone help me with this one?