#OK to post homework
#Tianyi Liu, Oct 11, Assignment 9
1.
(i) 15151926156281286723385293406631432927528001959338425859024498452242994993834111
(ii) 0

2.
WordsMod:=proc(a,S,x) local f:

if a=0 then
  print('bad input'):
  RETURN(FAIL):
fi:

f:=add(x^(a-i)/(1-x^a),i=1..a-1):

1/(1-f):
end:

#of ways without multiples of 5:118272656870560296827634782280

3.


4.
(i)[{(-n-2)*f(n+2)+(6+4*n)*f(n+1), f(0) = 1, f(1) = 2}, ogf]
A000984
(ii)[{(-n-3)*f(n+3)+(15+6*n)*f(n+2)+(-n-2)*f(n+1), f(0) = 1, f(1) = 3, f(2) = 13}, ogf]
A001850
(iii)[{(352*n^3+2768*n^2+7088*n+5920)*f(n+1)+(-484*n^3-4048*n^2-11184*n-10204)*f(n+2)+(-352*n^3-3120*n^2-9114*n-8766)*f(n+3)+(55*n^3+515*n^2+1570*n+1560)*f(n+4), f(0) = 1, f(1) = 2, f(2) = 14, f(3) = 84}, ogf]
A036692
(iv) Fail
A192365

5.
(i)[{(-27*n^2-81*n-60)*f(n+1)+(n^2+4*n+4)*f(n+2), f(0) = 1, f(1) = 6}, ogf]
A006480
(ii)[{(-27*n^3-216*n^2-537*n-420)*f(n+1)+(-108*n^3-972*n^2-2856*n-2760)*f(n+2)+(-107*n^3-1070*n^2-3555*n-3924)*f(n+3)+(2*n^3+22*n^2+80*n+96)*f(n+4), f(0) = 1, f(1) = 12, f(2) = 366, f(3) = 13800}, ogf]
A268550
(iii)[{(3*n^3+23*n^2+56*n+44)*f(n+1)+(-9*n^3-78*n^2-221*n-206)*f(n+2)+(171*n^3+1653*n^2+5281*n+5570)*f(n+3)+(-3*n^3-32*n^2-112*n-128)*f(n+4), f(0) = 1, f(1) = 13, f(2) = 409, f(3) = 16081}, ogf]
A126086

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),x,1i),x2=0,N+1),x2,i),x3=0,N+1),x3,i),x4=0,N+1),x4,i),i=0..N)]:

end:

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: