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)
2
n
Out[]=
SeriesCoefficient[(z+1)^r/(z-1)^(r+1),{z,0,d}]
In[]:=
|
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[]=
%/.d3
In[]:=
1
3
2
r
3
r
Out[]=
Binomial[r,d]Hypergeometric2F1[-d,1+r,1-d+r,-1]/.dLog[2,3]
In[]:=
Binomialr,Hypergeometric2F11+r,-,1+r-,-1
Log[3]
Log[2]
Log[3]
Log[2]
Log[3]
Log[2]
Out[]=
FullSimplify[%]
In[]:=
Binomialr,Hypergeometric2F11+r,-,r-,-1
Log[3]
Log[2]
Log[3]
Log[2]
2ArcCoth[5]
Log[2]
Out[]=
Plot[%,{r,0,10}]
In[]:=
Out[]=
Series[%122,{r,Infinity,2}]
In[]:=
Out[]=
(-1)^k/k!Pochhammer[-d,k]
Leading term
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+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[]=
Sumr^(d-k),{k,0,d}
d-k
2
Pochhammer[2,-1+d-k]
In[]:=
2r
Gamma[1+d]
Out[]=
%/.d3
In[]:=
1
6
2r
Out[]=
FunctionExpand[%]
In[]:=
12r+24+24+16
2
r
3
r
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[]=
(-d,r+1;-d+r+1;-1)
r |
d |
2
F
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,10}
d
2
d!
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
d!
Out[]=
Sum[2^k/k!r^k,{k,0,d}]
In[]:=
2r
Gamma[1+d]
Out[]=
Table[%,{d,1,6}]
In[]:=
Gamma[2,2r],Gamma[3,2r],Gamma[4,2r],Gamma[5,2r],Gamma[6,2r],Gamma[7,2r]
2r
1
2
2r
1
6
2r
1
24
2r
1
120
2r
1
720
2r
Out[]=
Expand[FunctionExpand[%]]
In[]:=
1+2r,1+2r+2,1+2r+2+,1+2r+2++,1+2r+2+++,1+2r+2++++
2
r
2
r
4
3
r
3
2
r
4
3
r
3
2
4
r
3
2
r
4
3
r
3
2
4
r
3
4
5
r
15
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++2+
8r
3
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+3
2
d
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
Next try
Leading term
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
3Pochhammer[1,d]
Out[]=
FunctionExpand[%]
In[]:=
d
2
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[]:=
$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,,,,,,,,,,,,,,,,,0,0,0,0,0,0,0,0,0,0,0,0
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
Out[]=
FindSequenceFunction2,2,,,,,,,,,,,,,,,,
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
3Gamma[1+n]
In[]:=
-2+d
2
3Gamma[-1+d]
Out[]=
FunctionExpand[%]
In[]:=
-2+d
2
3Gamma[-1+d]
Out[]=
FullSimplify[%]
In[]:=
-2+d
2
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+,++,++,++
2
r
8r
3
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[]=