In[]:=
(ReplaceAll[n3])[(Sum[a[k]*a[n-k+1]*k*((n-3*k)+2),​​{k,2,n-1}]+Sum[Sum[a[k]*a[m]*a[n-k-m+1]*k*(2*k-1),​​{m,1,n-k}],{k,1,n-1}])/((2*n-1)*(n-1))]
Out[]=
11
45
In[]:=
a[1]=1,a[2]=13,a[3]=1145,a[4]=73315,a[5]=354814175,a[6]=
136883
467775
,a[7]=
15360178
42567525
,a[8]=
26838347
58046625
,a[9]=
59574521252
97692469875
,a[10]=
7635110149532
9280784638125

Out[]=
1,
1
3
,
11
45
,
73
315
,
3548
14175
,
136883
467775
,
15360178
42567525
,
26838347
58046625
,
59574521252
97692469875
,
7635110149532
9280784638125

In[]:=
NumberForm[{FindRoot[​​x-Cos[x],{x,1}],ArcSin[1-2Sum[a[n](π/8.0)^(2n),{n,1,10}]]},7]
Out[]//NumberForm=
{{x0.7390851},0.7390851}
In[]:=
ReplaceAll[n8][1/(3n(2n+1))Sum[Sum[b[k]b[m]b[n-k-m-1](2(2m+1)(2n-2k-2m-1)-k(2k+1)),{m,0,n-k-1}],{k,0,n-1}]]
Out[]=
59574521252
71217810538875
In[]:=
b[0]=1,b[1]=
2
9
,b[2]=
11
135
,b[3]=
292
8505
,b[4]=
3548
229635
,b[5]=
273766
37889775
,b[6]=
15360178
4433103675
,b[7]=
214706776
126947968875
,b[8]=
59574521252
71217810538875
,b[9]=
15270220299064
36534736806442875

Out[]=
1,
2
9
,
11
135
,
292
8505
,
3548
229635
,
273766
37889775
,
15360178
4433103675
,
214706776
126947968875
,
59574521252
71217810538875
,
15270220299064
36534736806442875

In[]:=
NumberForm[{FindRoot[​​x-Cos[x],{x,1}],-Tanh[2ArcTanh[1/Sqrt[3](-Sum[b[n](Sqrt[3]π/8.0)^(2n+1),{n,0,9}])]]},7]
Out[]//NumberForm=
{{x0.7390851},0.7390851}