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

Leading term

Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^d],{d,10}]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175

Out[]=
FindSequenceFunction[%,d]
In[]:=
d
2
Pochhammer[2,-1+d]
Out[]=
Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-1)],{d,10}]
In[]:=
Coefficient
:1 is not a valid variable.
Coefficient[1+2r,1],2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

Out[]=
Rest[%]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

Out[]=
FindSequenceFunction[%,d]
In[]:=
d
2
Pochhammer[2,-1+d]
Out[]=
Sum
d-k
2
Pochhammer[2,-1+d-k]
r^(d-k),{k,0,d}
In[]:=
2r

Gamma[1+d,2r]
Gamma[1+d]
Out[]=
%/.d3
In[]:=
1
6
2r

Gamma[4,2r]
Out[]=
FunctionExpand[%]
In[]:=
12r+24
2
r
+24
3
r
+16
4
r
12r
Out[]=
Expand[%]
In[]:=
1+2r+2
2
r
+
4
3
r
3
Out[]=
FullSimplify[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],d>0]
In[]:=
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]
Out[]=
FunctionExpand[%,d>0]
In[]:=
Gamma[1+r]Hypergeometric2F1[-d,1+r,1-d+r,-1]
Gamma[1+d]Gamma[1-d+r]
Out[]=

r
d

2
F
1
(-d,r+1;-d+r+1;-1)
2r

Γ(d+1,2r)
Γ(d+1)
FunctionExpand
d
2
Pochhammer[2,-1+d]

In[]:=
d
2
Gamma[1+d]
Out[]=
Gamma[5]
In[]:=
24
Out[]=
4!
In[]:=
24
Out[]=
2^d/d!
In[]:=
Table
d
2
d!
,{d,10}
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175

Out[]=
Sum[2^(d-k)/(d-k)!r^(d-k),{k,0,d}]
In[]:=
2r

Gamma[1+d,2r]
d!
Out[]=
Sum[2^k/k!r^k,{k,0,d}]
In[]:=
2r

Gamma[1+d,2r]
Gamma[1+d]
Out[]=
Table[%,{d,1,6}]
In[]:=

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]
Out[]=
Expand[FunctionExpand[%]]
In[]:=
1+2r,1+2r+2
2
r
,1+2r+2
2
r
+
4
3
r
3
,1+2r+2
2
r
+
4
3
r
3
+
2
4
r
3
,1+2r+2
2
r
+
4
3
r
3
+
2
4
r
3
+
4
5
r
15
,1+2r+2
2
r
+
4
3
r
3
+
2
4
r
3
+
4
5
r
15
+
4
6
r
45

Out[]=
1/3(3+8r+6r^2+4r^3)//Expand
In[]:=
1+
8r
3
+2
2
r
+
4
3
r
3
Out[]=
Table[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],{d,6}]
In[]:=
Out[]=
FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],r>0]
In[]:=
Gamma[1+r]Hypergeometric2F1[-d,1+r,1-d+r,-1]
Gamma[1+d]Gamma[1-d+r]
Out[]=
Series[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],{r,Infinity,2}]
In[]:=
Out[]=
Series[r^-dBinomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],{r,Infinity,2}]
In[]:=
Hypergeometric2F1[-d,1+r,1-d+r,-1]
1
dGamma[d]
+
1-d
2Gamma[d]r
+
-2+9d-10
2
d
+3
3
d
24Gamma[d]
2
r
+
5/2
O
1
r

Out[]=
Series[r^-dBinomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1],{r,0,2}]
In[]:=
Out[]=
FullSimplify[%,d>0]
In[]:=
Out[]=

Next try

Leading term

Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^d],{d,10}]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175

Out[]=
FindSequenceFunction[%,d]
In[]:=
d
2
Pochhammer[2,-1+d]
Out[]=
FunctionExpand
d
2
Pochhammer[2,-1+d]

In[]:=
d
2
Gamma[1+d]
Out[]=
Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-1)],{d,2,10}]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

Out[]=
Rest[%]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835

Out[]=
Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-2)],{d,3,10}]
In[]:=

8
3
,
10
3
,
8
3
,
14
9
,
32
45
,
4
15
,
16
189
,
22
945

Out[]=
FindSequenceFunction[%,d]
In[]:=
d
2
(3+d)
3Pochhammer[1,d]
Out[]=
FunctionExpand[%]
In[]:=
d
2
(3+d)
3Gamma[1+d]
Out[]=
ParallelTable[FunctionExpand[FindSequenceFunction[Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-k)],{d,k+1,30}]]],{k,0,10}]
In[]:=
LaunchKernels
:Kernel KernelObject
Name: \delta
 resurrected as KernelObject
Name: \delta
KernelID: \65
.
LaunchKernels
:Kernel KernelObject
Name: \delta
 resurrected as KernelObject
Name: \delta
KernelID: \66
.
LaunchKernels
:Kernel KernelObject
Name: \delta
 resurrected as KernelObject
Name: \delta
KernelID: \67
.
General
:Further output of LaunchKernels::clone will be suppressed during this calculation.
$Aborted
Out[]=
ParallelTable[FunctionExpand[FindSequenceFunction[Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-k)],{d,k+1,30}]]],{k,0,3}]
In[]:=
With[{k=0},Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-k)],{d,k+1,30}]]
In[]:=
2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175
,
8
155925
,
4
467775
,
8
6081075
,
8
42567525
,
16
638512875
,
2
638512875
,
4
10854718875
,
4
97692469875
,0,0,0,0,0,0,0,0,0,0,0,0
Out[]=
FindSequenceFunction2,2,
4
3
,
2
3
,
4
15
,
4
45
,
8
315
,
2
315
,
4
2835
,
4
14175
,
8
155925
,
4
467775
,
8
6081075
,
8
42567525
,
16
638512875
,
2
638512875
,
4
10854718875
,
4
97692469875

In[]:=
#1
2
Pochhammer[2,-1+#1]
&
Out[]=
ParallelTable[Table[Coefficient[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],r^(d-k)],{d,k+1,30}],{k,0,10}]
In[]:=
Out[]=
DeleteCases[#,0]&/@%
In[]:=
Out[]=
FindSequenceFunction[#,n]&/@%
In[]:=
Out[]=
FunctionExpand[%]
In[]:=
Out[]=
2^(d-1)/(d-1)!
n
2
(3+n)
3Gamma[1+n]
/.nd-2
In[]:=
-2+d
2
(1+d)
3Gamma[-1+d]
Out[]=
FunctionExpand[%]
In[]:=
-2+d
2
(1+d)
3Gamma[-1+d]
Out[]=
FullSimplify[%]
In[]:=
-2+d
2
(1+d)
3Gamma[-1+d]
Out[]=
Table[2^d/d!r^d+2^(d-1)/(d-1)!r^(d-1)+2^(d-2)(1+d)/(3(d-2)!)r^(d-2),{d,1,6}]
In[]:=
1+2r,1+2r+2
2
r
,
8r
3
+2
2
r
+
4
3
r
3
,
10
2
r
3
+
4
3
r
3
+
2
4
r
3
,
8
3
r
3
+
2
4
r
3
+
4
5
r
15
,
14
4
r
9
+
4
5
r
15
+
4
6
r
45

Out[]=
Table[FunctionExpand[Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]],{d,1,6}]
In[]:=
Out[]=