It is not possible to get a series with $\{-1,0,1\}$ coefficients this way. The maximum one can get is having such coefficients for degrees up to 121. Here is one example that achieves this many $\{-1,0,1\}$ coefficients:
Numerator:
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 - x^8 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14 - x^15 - x^16 - x^17 - x^18 - x^19 - x^20 - x^21 - x^22 - x^23 - x^24 - x^25 - x^26 - x^27 - x^28 - x^29 - x^30 - x^31 + x^32 + x^33 + x^34 + x^35 + x^36 + x^37 + x^38 + x^39 + x^40 + x^41 + x^42 + x^43 + x^44 + x^45 + x^46 + x^47 + x^48 + x^49 + x^50 + x^51 + x^52 + x^53 + x^54 + x^55 + x^56 + x^57 + x^58 - x^59 + x^60 - x^61 + x^62 - x^63 - x^64 - x^65 - x^66 + x^67 + x^68 - x^69 + x^70 + x^71 + x^72 + x^73 + x^74 + x^75 + x^76 + x^77 + x^78 - x^79 - x^80 - x^81 - x^82 - x^83 - x^84 - x^85 + x^86 + x^87 - x^88 + x^89 + x^90 - x^91 + x^92 - x^93 - x^94 - x^95 + x^96 - x^97 - x^98 - x^99 + x^100 + x^101 - x^102 + x^103 + x^104 + x^105 - x^106 - x^107 + x^108 - x^109 + x^110 + x^111 - x^112 - x^113 + x^114 - x^115 + x^116 - x^117 - x^118 + x^119 - x^120 + x^121 + O(x^122)
Denominator:
(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*(1+x^6)*(1-x^7)*(1+x^8)*(1-x^9)*(1+x^10)*(1+x^11)*(1+x^12)*(1+x^13)*(1+x^14)*(1-x^15)*(1+x^16)*(1-x^17)*(1+x^18)*(1+x^19)*(1+x^20)*(1+x^21)*(1+x^22)*(1-x^23)*(1+x^24)*(1-x^25)*(1+x^26)*(1+x^27)*(1+x^28)*(1+x^29)*(1+x^30)*(1-x^31)*(1+x^32)*(1-x^33)*(1+x^34)*(1+x^35)*(1+x^36)*(1+x^37)*(1+x^38)*(1-x^39)*(1+x^40)*(1-x^41)*(1+x^42)*(1+x^43)*(1+x^44)*(1+x^45)*(1+x^46)*(1-x^47)*(1+x^48)*(1-x^49)*(1+x^50)*(1+x^51)*(1+x^52)*(1+x^53)*(1+x^54)*(1-x^55)*(1+x^56)*(1-x^57)*(1+x^58)*(1-x^59)*(1+x^60)*(1+x^61)*(1+x^62)*(1-x^63)*(1+x^64)*(1-x^65)*(1+x^66)*(1+x^67)*(1+x^68)*(1+x^69)*(1+x^70)*(1-x^71)*(1+x^72)*(1-x^73)*(1+x^74)*(1+x^75)*(1+x^76)*(1+x^77)*(1+x^78)*(1-x^79)*(1+x^80)*(1-x^81)*(1+x^82)*(1-x^83)*(1+x^84)*(1+x^85)*(1+x^86)*(1-x^87)*(1-x^88)*(1+x^89)*(1+x^90)*(1-x^91)*(1+x^92)*(1+x^93)*(1+x^94)*(1-x^95)*(1-x^96)*(1+x^97)*(1+x^98)*(1-x^99)*(1+x^100)*(1+x^101)*(1-x^102)*(1+x^103)*(1-x^104)*(1+x^105)*(1-x^106)*(1-x^107)*(1+x^108)*(1+x^109)*(1-x^110)*(1-x^111)*(1-x^112)*(1+x^113)*(1-x^114)*(1+x^115)*(1-x^116)*(1+x^117)*(1-x^118)*(1-x^119)*(1+x^120)*(1-x^121) + O(x^122)
Resulting series:
1 + x - x^3 - x^4 - x^5 - x^6 + x^7 - x^12 - x^13 + x^14 + x^16 + x^17 + x^18 - x^20 + x^23 + x^24 - x^29 + x^32 + x^33 - x^35 - x^36 - x^37 - x^38 + x^39 + x^41 - x^43 - x^44 + x^48 + x^49 - x^51 - x^52 - x^53 - x^54 + x^56 + x^60 - x^61 - x^63 + x^67 - x^68 + x^69 + x^71 + x^72 + x^73 - x^76 + x^77 + x^81 + x^82 + x^83 - x^85 + x^87 - x^88 - x^89 + x^90 - x^91 + x^92 - x^93 - x^95 - x^99 - x^102 - x^104 + x^105 + x^107 - x^111 - x^114 + x^115 + x^118 - x^119 + x^121 + O(x^122)