In[]:=
Sum[Binomial[n,k]Binomial[2^k+2^(n-k)-1,j](-1)^(j+1)/(​​j+1),{j,0,∞},{k,0,n}]​​  Block[{p=1/2,n=11},​​  Print[N@Table[Sum[Binomial[n,k]Binomial[2^k+2^(n-k)-1,j](-1)^(j+1)/(​​j+1),{j,0,Inf},{k,0,n}],{Inf,5,30,5}]];​​  ];
Out[]=
∞
∑
j=0
n
∑
k=0
1+j
(-1)
Binomial-1+
k
2
+
-k+n
2
,jBinomial[n,k]
1+j
{1.393×
14
10
,-6.39529×
25
10
,4.21479×
36
10
,-5.94834×
46
10
,2.56591×
56
10
,-4.23248×
65
10
}