#OK to post
# Soham Palande, Assignment 9, 10/11/2020


#PART 1

#(i)
RUID: 187006241

DiagSeq2(1/(1-x-(8*y)+(x*y)-(6*(x^3))+(2*(y^3))),x,y,50)


15191094275120822927061648250786876159006181233476515981823461429213801500829067921



#(ii)

DiagWalks3D({[1,7,1],[8,0,0],[0,7,7]},30)
[1, 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]


# PART 2





# PART 5

#(i)
#This sequence is in the OEIS
#A006480


#(ii)
#Not in OEIS

#(iii)
#Not in OEIS



#PART 6

#DiagWalks4D(S,N): Inputs  a set of "atomic steps" in the 2-dimenional square lattice, with positve steps, of the form [a,b,c,d]
#Number of walks from [0,0,0,0] to [n,n,n,n] using the atomic steps. 

DiagWalks4D:=proc(S,N) local s,x,y,z,w:
if not (type(S, set) and type(N,integer) and N>=0) then
 print(`Bad input`):
 RETURN(FAIL):
fi:
DiagSeq4(1/(1-add(x^s[1]*y^s[2]*z^s[3]*w^s[4], s in S)),x,y,z,w,N  ) :
end:




#DiagSeq4(f,x,y,z,w,N): Given a rational function f of the variables x, y,  z and w such that the denominator does not vanish at (0,0,0,0)
#outputs the list whose i-th entry is the coefficient of x^(i-1)*y^(i-1)*z^(i-1)*w(i-1) in the 4-variable power-series expansion of f
#Try:
#DiagSeq4(1/(1-x-y-z-w),x,y,z);
DiagSeq4:=proc(f,x,y,z,w,N) local i:

if subs({x=0,y=0,z=0,w=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,x=0,N+1),x,i),y=0,N+1),y,i),z=0,N+1),z,i),w=0,N+1),y,i),i=0..N)]:

end: