#OK to post homework
#Blair Seidler, 2021-03-14 Assignment 15

with(combinat):

Help:=proc():
print(`UDbetter(n),PiGuillera(N)`):
end:

# 1.
#UDbetter(n): The set of up-down permutations of length n, using the ideas behind
# the enumeration procedure ud(n) done in class
UDbetter:=proc(n) local numleft,Left,Right,L,R,l,r,G:
option remember:
if n=0 then RETURN({[]}): fi:
if n=1 then RETURN({[1]}): fi:
G:={}:
for numleft from 1 to n-1 by 2 do
  L:=UDbetter(numleft):
  R:=UDbetter(n-numleft-1):
  for Left in choose(n-1,numleft) do
    Right:=convert({seq(1..n-1)} minus convert(Left,set),list):
    G:=G union {seq(seq([op(l),n,op(r)],
	l in subs({seq(i=Left[i],i=1..nops(Left))},L)),
        r in subs({seq(i=Right[i],i=1..nops(Right))},R))}
  od:
od:
G:
end:


# 2. Not getting this one done by the due date. I will think about it some more


# 3.
#PiGuillera(N): Outputs an approximation to pi equal to the sum of the
#first N terms of Jesus Guillera's series.
PiGuillera:=proc(N) local sum,n:
sum:=0:
for n from 0 to N do
  sum:=sum+(-1)^n*binomial(2*n,n)^5*(820*n^2+180*n+13)/2^(20*n):
od:
sqrt(128/sum):
end:

#### Setting Digits:=10000, PiGuillera(3340)-Pi returns 0.