1. 
(i) 0 

(ii) 4741685087960650685822461715671211099459856575685906645393

(iii) 234303065886413369536085349033389448858104244347193374824719

2. 
Pnk:=proc(n,k) local r,s,t,i:
option remember:

if n < 0 or k < 0 then 
 RETURN():
fi:

if n = 0 and k = 0 then
 return {[]}:
fi:

if k = 0 or k > n then
 return {}:
fi:

r := {}:
s := {}:

for i from 0 to k do
  s := s union Pnk(n-k,i):
od:

r:= {seq([k,op(t)], t in s)}:
r:

end:

3. 
pnk:=proc(n,k) local r, i:
option remember:

if n < 0 or k < 0 then 
 RETURN():
fi:

if n = 0 and k = 0 then
 return 1:
fi:

if k = 0 or k > n then
 return 0:
fi:

r := 0:

for i from 0 to k do
  r := r + pnk(n-k,i):
od:

r:

end:


4.
pn:=proc(n) local t, i:
option remember:

if n <= 0 then 
 RETURN():
fi:

t := 0:

for i from 0 to n do
  t := t + pnk(n,i):
od:

t:

end:

seq(pn(n),n=0...30) = [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604]

--> A41 in the OEIS



5. 
[seq(p(5*n+4) mod 5, n=1..50)] = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

[seq(p(7*n+5) mod 7, n=1..50)] = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

[seq(p(11*n+6) mod 11, n=1..50)] = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]