(*FirstsketchyProof*)
In[]:=
HoldFormSum[Binomial[n-1,n-i]*Sum[Binomial[k+i,i]*Binomial[n-1,n-k],{k,0,n}],{i,0,n}]==Sum[Binomial[n-1,n-i]*(1+i)Hypergeometric2F1[2+i,1-n,2,-1],{i,0,n}]==IntegrateSumBinomial[n-1,n-i]*(1+i)*
n
x
i
(1+x)
-2-i
(2+x)
Gamma[1-n]Gamma[1+n]
,{i,0,n},{x,0,Infinity}==IntegrateSum
(1+i)
n
x
i
(1+x)
-2-i
(2+x)
Gamma[n]
Gamma[i]Gamma[1-n]Gamma[1+n]Gamma[1-i+n]
,{i,0,n},{x,0,Infinity}==
Gamma[n]
Gamma[1-n]Gamma[1+n]
*IntegrateSum
(1+i)
n
x
i
(1+x)
-2-i
(2+x)
Gamma[i]*Gamma[1-i+n]
,{i,0,n},{x,0,Infinity}==
Gamma[n]
Gamma[1-n]Gamma[1+n]
*Integrate
n
x
(1+x)
-2-n
(2+x)
-2+n
(3+2x)
(5+n+3x+nx)
Gamma[n]
,{x,0,Infinity}==
1
Gamma[1-n]Gamma[1+n]
*Integrate[
n
x
(1+x)
-2-n
(2+x)
-2+n
(3+2x)
(5+n+3x+nx),{x,0,Infinity}]==
1
(-1+2n)Gamma[1+n]Gamma[2+n]
-5+2n
2
Gamma[2n]8n(-3+5n+2
2
n
)Hypergeometric2F11-n,2-n,-2n,
3
4
-(-1+n)4(-4+7n+2
2
n
)Hypergeometric2F12-n,2-n,1-2n,
3
4
-(-10+3n+
2
n
)Hypergeometric2F12-n,3-n,2-2n,
3
4
+
1
Gamma[1-n]Gamma[2-n]
-5-2n
2
-1+2n
3
Gamma[-1-2n]8n(5+11n+2
2
n
)Hypergeometric2F11+n,2+n,2n,
3
4
-3(1+n)4(4+9n+2
2
n
)Hypergeometric2F12+n,2+n,1+2n,
3
4
-3(6+5n+
2
n
)Hypergeometric2F12+n,3+n,2(1+n),
3
4
==
1
9
π
-4(1+n)
2
1+8n
2
Gamma
1
2
+n2(66+n(31+n(10+n)))Hypergeometric2F11-n,2-n,-2n,
3
4
+(-1+n)(12+n(5+n))Hypergeometric2F12-n,2-n,-2n,
3
4

(1+n)Gamma[3+n]
+
1+2n
3
Gamma-
1
2
-n-(10+n(69+n(37+6n)))Hypergeometric2F11+n,2+n,2n,
3
4
+n(13+n(11+2n))Hypergeometric2F12+n,2+n,2n,
3
4

Gamma[3-n]
==
n
(-1)
-2+n
4
(-1+n)(3+n)JacobiP-2+n,-2n,3,-
1
2
-4(7+n(4+n))JacobiP-1+n,-1-2n,2,-
1
2

3n(1+n)
//TraditionalForm
Out[]//TraditionalForm=
n
∑
i=0

n-1
n-i

n
∑
k=0

k+i
i

n-1
n-k

n
∑
i=0

n-1
n-i
(1+i)
2
F
1
(2+i,1-n;2;-1)
∞
∫
0
n
∑
i=0

n-1
n-i
(1+i)
n
x
i
(1+x)
-2-i
(2+x)

Γ(1-n)Γ(1+n)
x
∞
∫
0
n
∑
i=0
(1+i)
n
x
i
(1+x)
-2-i
(2+x)
Γ(n)
Γ(i)Γ(1-n)Γ(1+n)Γ(1-i+n)
x
Γ(n)
∞
∫
0
n
∑
i=0
(1+i)
n
x
i
(1+x)
-2-i
(2+x)
Γ(i)Γ(1-i+n)
x
Γ(1-n)Γ(1+n)

Γ(n)
∞
∫
0
n
x
(1+x)
-2-n
(2+x)
-2+n
(3+2x)
(5+n+3x+nx)
Γ(n)
x
Γ(1-n)Γ(1+n)

∞
∫
0
n
x
(1+x)
-2-n
(2+x)
-2+n
(3+2x)
(5+n+3x+nx)x
Γ(1-n)Γ(1+n)

-5+2n
2
Γ(2n)8n(-3+5n+2
2
n
)
2
F
1
1-n,2-n;-2n;
3
4
-(-1+n)4(-4+7n+2
2
n
)
2
F
1
2-n,2-n;1-2n;
3
4
-(-10+3n+
2
n
)
2
F
1
2-n,3-n;2-2n;
3
4

(-1+2n)Γ(1+n)Γ(2+n)
+
-5-2n
2
-1+2n
3
Γ(-1-2n)8n(5+11n+2
2
n
)
2
F
1
1+n,2+n;2n;
3
4
-3(1+n)4(4+9n+2
2
n
)
2
F
1
2+n,2+n;1+2n;
3
4
-3(6+5n+
2
n
)
2
F
1
2+n,3+n;2(1+n);
3
4

Γ(1-n)Γ(2-n)

-4(1+n)
2
1+8n
2
Γ
1
2
+n2(66+n(31+n(10+n)))
2
F
1
1-n,2-n;-2n;
3
4
+(-1+n)(12+n(5+n))
2
F
1
2-n,2-n;-2n;
3
4
(1+n)Γ(3+n)
+
1+2n
3
Γ-
1
2
-n-(10+n(69+n(37+6n)))
2
F
1
1+n,2+n;2n;
3
4
+n(13+n(11+2n))
2
F
1
2+n,2+n;2n;
3
4
Γ(3-n)
9
π

n
(-1)
-2+n
4
(-1+n)(3+n)
(-2n,3)
P
-2+n
-
1
2
-4(7+n(4+n))
(-1-2n,2)
P
-1+n
-
1
2

3n(1+n)
(*SecondProof*)
In[]:=
A[n_]:=
1
3n(1+n)
n
(-1)
-2+n
4
(-1+n)(3+n)JacobiP-2+n,-2n,3,-
1
2
-4(7+n(4+n))JacobiP-1+n,-1-2n,2,-
1
2
;
In[]:=
a[n_]:=A[n-1];
(*From"https://oeis.org/"A051708->FORMULA->D-finitewithrecurrencea(1)=1;a(2)=2;a(n)=((10*n-16)*a(n-1)-(9*n-27)*a(n-2))/(n-1),forn>=3.-MartinJ.Erickson(erickson(AT)truman.edu),Nov12×2007*)
FullSimplifya[n]==
(10n-16)*a[n-1]-(9n-27)*a[n-2]
n-1

Out[]=
True