1.
N^2 - N - 1
2


3. 

S6:=proc(n) local k,sum: 
  sum:=0;
  for k from 0 to n do 
    sum:= sum + (binomial(n,k))^6:
  od:
  sum:
end:

S6seq:=proc(N) local i,s:
  s:={};
  for i from 0 to N do
    s:={op(s),S6(i)}:
  od:
  s:
end:

time(S6seq(1000)) = 7.562

{1, 2, 66, 1460, 54850, 2031252, 86874564, 3848298792, 180295263810, 8709958973540, 433617084579316, 22071658807720392, ...}


S6seqClever:=proc(N) local i,s:
  s:={};
  for i from 0 to N do
    s:={op(s),SeqFromRec(N^6,n,N,[1, 2, 66, 1460, 54850, 2031252], N)}:
  od:
  s:
end: