I tried several different ways with things like this:

Sum(Binomial(n-1,n-i)*Sum(Binomial(k+i,i)*Binomial(n-1,n-k),{k,0,n}),{i,0,n})

Unfortunately, none of those succeeded.

I finally used brute force calculation for n=0,n=1,...n=8 to get

{1, 2, 14, 106, 838, 6802, 56190, 470010, 3968310}

And I asked OEIS for all the information they had on this.

OEIS Link

which gives a collection of interesting places where this has been seen, information about the sequence, formulas to calculate it, etc.

Please check all this very carefully to find any of my mistakes before you even think about depending on this.

g[n_] := Sum[Binomial[n - 1, n - i]*
Sum[Binomial[k + i, i]*Binomial[n - 1, n - k], {k, 0, n}], {i, 0, 
n}];
Table[g[n], {n, 1, 15}]
(* {2, 14, 106, 838, 6802, 56190, 470010, 3968310, 33747490, 288654574, \
2480593546, 21400729382, 185239360178, 1607913963614, 13991107041306}*)

A051708 or Closed Form solution for the series works for: $n\in \mathbb{Z}$ and $n\geq 1$:

 W[n_] :=((-1)^n 4^(-2 + 
   n) ((-1 + n) (3 + n) JacobiP[-2 + n, -2 n, 3, -(1/2)] - 4 (7 + n (4 + n)) JacobiP[-1 + n, -1 - 2 n, 2, -(1/2)]))/(3 n (1 + n));
 Table[W[n], {n, 1, 15}]

 (*{2, 14, 106, 838, 6802, 56190, 470010, 3968310, 33747490, 288654574, \
 2480593546, 21400729382, 185239360178, 1607913963614, 13991107041306}*)

We can check(Code is Slow):

  g[n_] := 
     Sum[Binomial[n - 1, n - i]*
       Sum[Binomial[k + i, i]*Binomial[n - 1, n - k], {k, 0, n}], {i, 0, 
       n}];

  g1[n_] := ((-1)^
    n 4^(-2 + 
     n) ((-1 + n) (3 + n) JacobiP[-2 + n, -2 n, 3, -(1/2)] - 
      4 (7 + n (4 + n)) JacobiP[-1 + n, -1 - 2 n, 2, -(1/2)]))/(
   3 n (1 + n));
   Table[g[n] - g1[n], {n, 1, 200}]

   (*{0,0, and 0...}*)

Regards M.I.

No, W[n] and W2[m,n] are not close approximations. W[n] and W2[m,n] are exact solution.

For $n\in \mathbb{Z}$ and $m\in \mathbb{Z}$ this values are removable singularity points that why I use approximate values to get a solution.

If you need a exact value then you need to use a Limit.

Table[Limit[W[n], n -> j], {j, 2, 5}]
(* {14, 106, 838, 6802} *)
Limit[W2[m, n], {m -> 5, n -> 9}](*nested limit *)
(*365696*)


FunctionSingularities[W[n], n]

(*(n >= -(1/2) && Sin[(-(1/2) - n) \[Pi]] == 0) || 
 n == 0 || (n >= 2 && Sin[(2 - n) \[Pi]] == 0) || (n <= -2 && 
   Sin[(2 + n) \[Pi]] == 0) || (n <= 1/2 && 
   Sin[(-(1/2) + n) \[Pi]] == 0) || Gamma[2 - n] == 0 || 
 Gamma[2 + n] == 0 || (Sin[(1 - 2 n) \[Pi]] == 0 && Re[n] >= 1/2) || 
 Sin[2 n \[Pi]] == 
  0 || (Re[n] >= 1 && Sin[(2 - 2 n) \[Pi]] == 0) || (Re[n] <= -(1/2) &&
    Sin[(1 + 2 n) \[Pi]] == 0) || (Sin[2 (1 + n) \[Pi]] == 0 && 
   1 + Re[n] <= 0)*)

Regards M.I.

Dear Mariusz, Is your submission for your formula in A035002 ready to be proposed for review?

<< OEIS Server: This sequence has not been edited or commented on for a week yet is not proposed for review. If it is ready for review, please visit https://oeis.org/draft/A035002 and click the button that reads "These changes are ready for review by an OEIS Editor.">>

Proofs.