I try to prove the following hypothesis $$\sum_{i=0}^k \frac{(-1)^i (n+i-1)(n+k-2)!}{i!(k-1)!(n-1)!}{{n+2k-i-1} \choose {2k}}=\frac {{n+k-1\choose k}{n-2+k\choose k}}{k+1}.$$ In the Maple:
gip:=proc (n,i) simplify(binomial(n+i-1,i)*binomial(n-1-1+i,i)/(i+1)) end proc;
l:=proc (n,k) simplify(sum('(-1)^i*(n+i-1)*(n+k-2)!*binomial(n+2*k-i-1,2*k)/(i!(k-i)!(n-1)!)','i' = 0 .. k)) end proc;
for j from 0 by 1 to 12 do print(l(3,j),gip(3,j),l(4,j),gip(4,j),l(5,j),gip(5,j),l(6,j),gip(6,j),l(7,j),gip(7,j)) end do;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
3, 3, 6, 6, 10, 10, 15, 15, 21, 21
6, 6, 20, 20, 50, 50, 105, 105, 196, 196
10, 10, 50, 50, 175, 175, 490, 490, 1176, 1176
15, 15, 105, 105, 490, 490, 1764, 1764, 5292, 5292
21, 21, 196, 196, 1176, 1176, 5292, 5292, 19404, 19404
28, 28, 336, 336, 2520, 2520, 13860, 13860, 60984, 60984
36, 36, 540, 540, 4950, 4950, 32670, 32670, 169884, 169884
45, 45, 825, 825, 9075, 9075, 70785, 70785, 429429, 429429
55, 55, 1210, 1210, 15730, 15730, 143143, 143143, 1002001, 1002001
66, 66, 1716, 1716, 26026, 26026, 273273, 273273, 2186184, 2186184
78, 78, 2366, 2366, 41405, 41405, 496860, 496860, 4504864, 4504864
91, 91, 3185, 3185, 63700, 63700, 866320, 866320, 8836464, 8836464
But $$\sum_{i=0}^k \frac{(-1)^i (n+i-1)(n+k-2)!}{i!(k-1)!(n-1)!}{{n+2k-i-1} \choose {2k}}=\sum_{i=0}^k \frac{(-1)^i (n+i-1)(n+k-2)!k}{i!k!(n-2)!(n-1)}{{n+2k-i-1} \choose {2k}}=\\={n-2+k\choose k} \sum_{i=0}^k \frac{(-1)^i (n+i-1)k}{i!(n-1)}{{n+2k-i-1} \choose {2k}}.$$
And I should prove that $\sum_{i=0}^k \frac{(-1)^i (n+i-1)k}{i!(n-1)}{{n+2k-i-1} \choose {2k}}=\frac {{n+k-1\choose k}}{k+1}.$
I tried use Mathematical induction method (about k) and Chu's Theorem. The second way was using formula ${r+k+1 \choose k}=\sum_{i=0}^k {r+i \choose i}.$ I cannot prove it. Maple prove that it is false:
l1:=proc (n,k) simplify(sum('(-1)^i*(n+i-1)kbinomial(n+2*k-i-1,2*k)/(i!*(n-1))','i' = 0 .. k)) end proc; gip1:=proc (n,i) simplify(binomial(n+i-1,i)/(i+1)) end proc;
for j from 0 by 1 to 12 do print(l1(3,j),gip1(3,j),l1(4,j),gip1(4,j),l1(5,j),gip1(5,j)) end do; 0, 1, 0, 1, 0, 1
3/2, 3/2, 2, 2, 5/2, 5/2
17, 2, 115/3, 10/3, 75, 5
111/2, 5/2, 313/2, 5, 2975/8, 35/4
130, 3, 1346/3, 7, 7679/6, 14
505/2, 7/2, 6205/6, 28/3, 27595/8, 21
435, 4, 2065, 12, 31579/4, 30
385679
1379/2, 9/2, 22351/6, 15, ------, 165/4
24
1028, 5, 18692/3, 55/3, 29949, 55
2925/2, 11/2, 19659/2, 22, 416847/8, 143/2
1029535
2005, 6, 44405/3, 26, -------, 91
12
5335/2, 13/2, 128755/6, 91/3, 1080233/8, 455/4
3462, 7, 30146, 35, 409327/2, 140
Where is my eror?
In the definition of l(n,k), why do you have $(k-i)!$ in the denominator instead of $k!$?
There is a line in the Maple code that defines $l$ as a procedure. Towards the end of the line, one of the factors is $(k-i)!$, in the denominator. None of the factors in the MathJax formula above it contain $(k-i)!$.