In[]:=
Table[TraditionalForm[Simplify[Abs[SeriesCoefficient[(z+1)^r/(z-1)^(r+1),{z,0,d}]],r>0]],{d,10}]
Out[]=
In[]:=
Simplify[%,n>0]
Out[]=
1-n
(-1)
(1+2n+2
2
n
)
In[]:=
SeriesCoefficient[(z+1)^r/(z-1)^(r+1),{z,0,d}]
Out[]=
1-r
(-1)
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]
d≥0
0
True
In[]:=
FullSimplify[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]]
Out[]=
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]
In[]:=
%/.d3
Out[]=
1
3
(3+8r+6
2
r
+4
3
r
)
In[]:=
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]/.dLog[2,3]
Out[]=
Binomialr,
Log[3]
Log[2]
Hypergeometric2F11+r,-
Log[3]
Log[2]
,1+r-
Log[3]
Log[2]
,-1
In[]:=
FullSimplify[%]
Out[]=
Binomialr,
Log[3]
Log[2]
Hypergeometric2F11+r,-
Log[3]
Log[2]
,r-
2ArcCoth[5]
Log[2]
,-1
In[]:=
Plot[%,{r,0,10}]
Out[]=
In[]:=
Series[%122,{r,Infinity,2}]
Out[]=
(-1)^k/k!Pochhammer[-d,k]

Leading term

In[]:=
Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^d],{d,10}]
Out[]=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175

In[]:=
FindSequenceFunction[%,d]
Out[]=
d
2
Pochhammer[2,-1+d]
In[]:=
Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-1)],{d,10}]
Coefficient
:1 is not a valid variable.
Out[]=
Coefficient[1+2r,1],2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

In[]:=
Rest[%]
Out[]=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

In[]:=
FindSequenceFunction[%,d]
Out[]=
d
2
Pochhammer[2,-1+d]
In[]:=
Sum
d-k
2
Pochhammer[2,-1+d-k]
r^(d-k),{k,0,d}
Out[]=
2r

Gamma[1+d,2r]
Gamma[1+d]
In[]:=
%/.d3
Out[]=
1
6
2r

Gamma[4,2r]
In[]:=
FunctionExpand[%]
Out[]=
12r+24
2
r
+24
3
r
+16
4
r
12r
In[]:=
Expand[%]
Out[]=
1+2r+2
2
r
+
4
3
r
3
In[]:=
FullSimplify[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],d>0]
Out[]=
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]
In[]:=
FunctionExpand[%,d>0]
Out[]=
Gamma[1+r]Hypergeometric2F1[-d,1+r,1-d+r,-1]
Gamma[1+d]Gamma[1-d+r]

r
d

2
F
1
(-d,r+1;-d+r+1;-1)
2r

Γ(d+1,2r)
Γ(d+1)
In[]:=
FunctionExpand
d
2
Pochhammer[2,-1+d]

Out[]=
d
2
Gamma[1+d]
In[]:=
Gamma[5]
Out[]=
24
In[]:=
4!
Out[]=
24
In[]:=
2^d/d!
In[]:=
Table
d
2
d!
,{d,10}
Out[]=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175

In[]:=
Sum[2^(d-k)/(d-k)!r^(d-k),{k,0,d}]
Out[]=
2r

Gamma[1+d,2r]
d!
In[]:=
Sum[2^k/k!r^k,{k,0,d}]
Out[]=
2r

Gamma[1+d,2r]
Gamma[1+d]
In[]:=
Table[%,{d,1,6}]
Out[]=

2r

Gamma[2,2r],
1
2
2r

Gamma[3,2r],
1
6
2r

Gamma[4,2r],
1
24
2r

Gamma[5,2r],
1
120
2r

Gamma[6,2r],
1
720
2r

Gamma[7,2r]

Next try

Leading term