#OK to post homework 
#Zhizhang Deng, 10/17/2020, Assignment 9

1. 
(i). Unable to calculate. Maple will complain. I tried other people's ID it works. but my ID doesn't work. 
This is my function: 1/(-7*x^3 + y^3 - x + xy - 7*y + 1)
(ii). 0

2. dont know how to do

3. dont know how to do

4. 
(i). [A000984] f(n) = (4* n - 2) * f(n - 1) / n [initial: f(0) = 1, f(1) = 2]
(ii). [A001850] f(n) = -(-f(n - 2) + 3 * f(n - 1) - (-f(n - 2) + 6 * f(n - 1)) * n) / n [initial: f(0) = 1, f(1) = 3, f(2) = 13]
(iii). [A036692] f(n) = (672*f(n - 3) - 740*f(-2 + n) - 298*f(-1 + n) - (1840*f(n - 3) - 2032*f(-2 + n) - 1050*f(-1 + n) + (-1456*f(n - 3) + 1760*f(-2 + n) + 1104*f(-1 + n) + (352*f(n - 3) - 484*f(-2 + n) - 352*f(-1 + n))*n)*n)*n)/((90 + (-145 + 55*n)*n)*n) [initial f(0)=1, f(1)=2, f(2)=14, f(3)=84
(iv). [A192365] FAIL

5. 
(i). [A006480] f(n) = -(-6 + (27 - 27*n)*n)*f(-1 + n) / n^2 [initial: f(0) = 1, f(1) = 6]

(ii). [A268542] f(n) = (-1680*f(-3 + n) - 1920*f(n - 2) - 288*f(-1 + n) - (-3933*f(-3 + n) - 5610*f(n - 2) - 1344*f(-1 + n) + (2700*f(-3 + n) + 4819*f(n - 2) + 1856*f(-1 + n) + (-567*f(-3 + n) - 1281*f(n - 2) - 672*f(-1 + n))*n)*n)*n)/((-74 + 42*n)*n^2) [initial: f(0) = 1, f(1) = 4, f(2) = 42, f(3) = 520]

(iii). [A112019] f(n) = (-96*f(-3 + n) - 320*f(n - 2) - 800*f(n - 1) - (-332*f(-3 + n) - 1065*f(n - 2) - 3797*f(n - 1) + (260*f(-3 + n) + 1005*f(n - 2) + 5538*f(n - 1) + (-59*f(-3 + n) - 295*f(n - 2) - 2301*f(n - 1))*n)*n)*n)/((-166 + 118*n)*n^2) [initial: f(0) = 1, f(1) = 5, f(2) = 55, f(3) = 749

6. 
DiagSeq4:=proc(f,x1,x2,x3,x4,N) local i:

if subs({x1=0,x2=0,x3=0,x4=0},denom(f))=0 then
 print(`The denominator of`, f, `should have a non-zero constant term `):
 RETURN(FAIL):
fi:

[seq(coeff(taylor(coeff(taylor(coeff(taylor(coeff(taylor(f,x1=0,N+1),x1,i),x2=0,N+1),x2,i),x3=0,N+1),x3,i),x4=0,N+1),x4,i),i=0..N)]:

end:

#DiagWalks3D(S,N): Inputs  a set of "atomic steps" in the 2-dimenional square lattice, with positve steps, of the form [a,b,c]
#Number of walks from [0,0,0] to [n,n,n] using the atomic steps. For example try:
#DiagWalks3D({[1,0,0],[0,1,0],[0,0,1]},40);
DiagWalks4D:=proc(S,N) local s,x1,x2,x3,x4:
if not (type(S, set) and type(N,integer) and N>=0) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

DiagSeq4(1/(1-add(x1^s[1]*x2^s[2]*x3^s[3]*x4^s[4], s in S)),x1,x2,x3,x4,N  ) :
end: