#OK to post
#Yuxuan Yang, Feb 18th, Assignment 8

with(combinat):


#1
SPnF:=proc(n) local i: SPnFs({seq(i,i=1..n)}): end:

SPnFs:=proc(S) local subs,s,i,sp,ans,s1,n,S1:
if nops(S)=0 then
RETURN({{}}):
fi:
ans:={}:
n:=S[nops(S)]:
S1:=S minus {n}:
subs:=powerset(S1):
for s in subs do
 sp:=SPnF(S1 minus s):
 for s1 in sp do
  ans:=ans union {s1 union {s union {n}}}:
 od:
od:
end:


#3
#seq(coeff(taylor(log(add(2^(n*(n - 1)/2)*x^n/n!, n = 0 .. 60)), x, 30), x, j)*j!, j = 1 .. 30)
#
#A001187 in OEIS


#5
NuCG:=proc(n,k) local i,j,f,x,a:
f:=log(add((1+a)^(m*(m-1)/2)*x^m/m!,m=0..3*n)):
coeff(taylor(coeff(taylor(f,x,n+1),x,n)*n!,a,k+1),a,k):
end:

#seq(NuCG(n,n-1),n=1..20)  A000272
#seq(NuCG(n,n),n=1..20)    A057500
#seq(NuCG(n,n+1),n=1..20)  A061540
#seq(NuCG(n,n+2),n=1..20)  A061541
#seq(NuCG(n,n+3),n=1..20)  A061542
#seq(NuCG(n,n+4),n=1..20)  A061543
#seq(NuCG(n,n+5),n=1..20)  A096117
#seq(NuCG(n,n+6),n=1..20)  A061544