The Generating Function for the Number of Ways of Making Change of n money-units in a County whose coins have denominatons, [1, 5, 10, 25, 50, 100] and where all coins (that show up) show up different number of times. By Shalosh B. Ekhad Prop.: Let a(n) be the number of ways of forming change of n money-units in a country where the coins have denominations [1, 5, 10, 25, 50, 100] and none of the coins that do show up at least once can occur the same number of times. Then infinity ----- 191 185 \ n 120 q q ) a(n) q = - -------- + ------------------------------------- / 191 5 35 150 ----- 1 - q (1 - q) (1 - q ) (1 - q ) (1 - q ) n = 0 185 q + ------------------------------------- 5 60 125 (1 - q) (1 - q ) (1 - q ) (1 - q ) 185 q + ------------------------------------- 5 110 75 (1 - q) (1 - q ) (1 - q ) (1 - q ) 180 q + -------------------------------------- 10 30 150 (1 - q) (1 - q ) (1 - q ) (1 - q ) 180 q + -------------------------------------- 10 55 125 (1 - q) (1 - q ) (1 - q ) (1 - q ) 180 q + -------------------------------------- 10 105 75 (1 - q) (1 - q ) (1 - q ) (1 - q ) 165 q + -------------------------------------- 25 15 150 (1 - q) (1 - q ) (1 - q ) (1 - q ) 165 q + -------------------------------------- 25 55 110 (1 - q) (1 - q ) (1 - q ) (1 - q ) 165 q + -------------------------------------- 25 105 60 (1 - q) (1 - q ) (1 - q ) (1 - q ) 140 q + -------------------------------------- 50 15 125 (1 - q) (1 - q ) (1 - q ) (1 - q ) 140 q + -------------------------------------- 50 30 110 (1 - q) (1 - q ) (1 - q ) (1 - q ) 140 q + -------------------------------------- 50 105 35 (1 - q) (1 - q ) (1 - q ) (1 - q ) 90 q + -------------------------------------- 100 15 75 (1 - q) (1 - q ) (1 - q ) (1 - q ) 90 q + -------------------------------------- 100 30 60 (1 - q) (1 - q ) (1 - q ) (1 - q ) 90 q + -------------------------------------- 100 55 35 (1 - q) (1 - q ) (1 - q ) (1 - q ) 176 q + --------------------------------------- 5 10 26 150 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 176 q + --------------------------------------- 5 10 51 125 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 176 q + --------------------------------------- 5 10 101 75 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 161 q + --------------------------------------- 5 25 11 150 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 161 q + --------------------------------------- 5 25 51 110 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 161 q + --------------------------------------- 5 25 101 60 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 136 q + --------------------------------------- 5 50 11 125 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 136 q + --------------------------------------- 5 50 26 110 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 136 q + --------------------------------------- 5 50 101 35 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 86 q + --------------------------------------- 5 100 11 75 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 86 q + --------------------------------------- 5 100 26 60 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 86 q + --------------------------------------- 5 100 51 35 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 156 q + --------------------------------------- 10 25 6 150 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 156 q + ---------------------------------------- 10 25 51 105 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 156 q + ---------------------------------------- 10 25 101 55 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 131 166 q 2 q + --------------------------------------- - ------------------------------ 10 50 6 125 25 55 111 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 166 166 2 q 2 q - ------------------------------ - ------------------------------ 25 105 61 25 101 65 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 166 185 2 q 6 q - ------------------------------ - --------------------------- 25 15 151 5 185 (1 - q ) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 180 165 6 q 6 q - ---------------------------- - ---------------------------- 10 180 25 165 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 140 90 6 q 6 q - ---------------------------- - ---------------------------- 50 140 100 90 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 190 190 2 q 2 q - ---------------------------- - ---------------------------- 15 175 30 160 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 190 190 2 q 2 q - ---------------------------- - ---------------------------- 55 135 105 85 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 190 190 2 q 2 q - ---------------------------- - ---------------------------- 35 155 60 130 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 190 190 2 q 2 q - ---------------------------- - ---------------------------- 110 80 75 115 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 190 190 2 q 2 q - ---------------------------- - ---------------------------- 125 65 150 40 (1 - q) (1 - q ) (1 - q ) (1 - q) (1 - q ) (1 - q ) 176 161 6 q 6 q - ----------------------------- - ----------------------------- 5 10 176 5 25 161 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 136 86 6 q 6 q - ----------------------------- - ----------------------------- 5 50 136 5 100 86 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 186 186 2 q 2 q - ----------------------------- - ----------------------------- 5 11 175 5 26 160 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 186 186 2 q 2 q - ----------------------------- - ----------------------------- 5 51 135 5 101 85 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 186 186 2 q 2 q - ----------------------------- - ----------------------------- 5 35 151 5 60 126 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 186 186 2 q 2 q - ----------------------------- - ----------------------------- 5 110 76 5 75 111 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 186 186 2 q 2 q - ----------------------------- - ----------------------------- 5 125 61 5 150 36 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 156 131 6 q 6 q - ------------------------------ - ------------------------------ 10 25 156 10 50 131 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 81 181 6 q 2 q - ------------------------------ - ----------------------------- 10 100 81 10 6 175 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 181 181 2 q 2 q - ------------------------------ - ------------------------------ 10 26 155 10 51 130 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 181 181 2 q 2 q - ------------------------------ - ------------------------------ 10 101 80 10 30 151 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 181 181 2 q 2 q - ------------------------------ - ------------------------------ 10 55 126 10 105 76 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 181 181 2 q 2 q - ------------------------------ - ------------------------------ 10 75 106 10 125 56 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 181 116 2 q 6 q - ------------------------------ - ------------------------------ 10 150 31 25 50 116 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 66 166 6 q 2 q - ------------------------------ - ----------------------------- 25 100 66 25 6 160 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 166 166 2 q 2 q - ------------------------------ - ------------------------------ 25 11 155 25 51 115 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 131 q + ---------------------------------------- 10 50 26 105 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 131 q + ---------------------------------------- 10 50 101 30 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 81 q + --------------------------------------- 10 100 6 75 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 81 q + ---------------------------------------- 10 100 26 55 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 81 q + ---------------------------------------- 10 100 51 30 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 116 q + --------------------------------------- 25 50 6 110 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 116 q + ---------------------------------------- 25 50 11 105 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 116 q + ---------------------------------------- 25 50 101 15 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 66 q + --------------------------------------- 25 100 6 60 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 66 q + ---------------------------------------- 25 100 11 55 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 66 q + ---------------------------------------- 25 100 51 15 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 41 q + --------------------------------------- 50 100 6 35 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 41 q + ---------------------------------------- 50 100 11 30 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 41 q + ---------------------------------------- 50 100 26 15 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 1 + --------------------------------------------------------- 5 10 25 50 100 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 190 186 181 24 q 24 q 24 q + ------------------ + ------------------- + -------------------- 190 5 186 10 181 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 166 141 91 24 q 24 q 24 q + -------------------- + -------------------- + -------------------- 25 166 50 141 100 91 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 6 q 6 q 6 q + ------------------- + -------------------- + -------------------- 6 185 11 180 26 165 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 6 q 6 q 6 q + -------------------- + -------------------- + -------------------- 51 140 101 90 15 176 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 6 q 6 q 6 q + -------------------- + -------------------- + -------------------- 30 161 55 136 105 86 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 6 q 6 q 6 q + -------------------- + -------------------- + -------------------- 35 156 60 131 110 81 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 6 q 6 q 6 q + -------------------- + -------------------- + -------------------- 75 116 125 66 150 41 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 4 q 4 q 4 q + -------------------- + -------------------- + -------------------- 16 175 31 160 56 135 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 4 q 4 q 4 q + -------------------- + -------------------- + -------------------- 106 85 36 155 61 130 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 191 4 q 4 q 4 q + -------------------- + -------------------- + -------------------- 111 80 76 115 126 65 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 166 4 q 2 q + -------------------- - ------------------------------ 151 40 25 60 106 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 166 166 2 q 2 q - ------------------------------ - ------------------------------ 25 110 56 25 150 16 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 41 141 6 q 2 q - ------------------------------ - ----------------------------- 50 100 41 50 6 135 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 141 141 2 q 2 q - ------------------------------ - ------------------------------ 50 11 130 50 26 115 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 141 141 2 q 2 q - ------------------------------ - ------------------------------ 50 101 40 50 15 126 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 141 141 2 q 2 q - ------------------------------ - ------------------------------ 50 30 111 50 105 36 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 141 141 2 q 2 q - ------------------------------ - ------------------------------ 50 35 106 50 110 31 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 141 91 2 q 2 q - ------------------------------ - ----------------------------- 50 125 16 100 6 85 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 91 91 2 q 2 q - ------------------------------ - ------------------------------ 100 11 80 100 26 65 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 91 91 2 q 2 q - ------------------------------ - ------------------------------ 100 51 40 100 15 76 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 91 91 2 q 2 q - ------------------------------ - ------------------------------ 100 30 61 100 55 36 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 91 91 2 q 2 q - ------------------------------ - ------------------------------ 100 35 56 100 60 31 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 91 191 2 q q - ------------------------------ - ----------------------------- 100 75 16 6 35 150 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ----------------------------- - ----------------------------- 6 60 125 6 110 75 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 11 30 150 11 55 125 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 11 105 75 26 15 150 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 26 55 110 26 105 60 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 51 15 125 51 30 110 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 51 105 35 101 15 75 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 191 191 q q - ------------------------------ - ------------------------------ 101 30 60 101 55 35 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 175 2 q + ------------------------------------- 5 10 175 (1 - q) (1 - q ) (1 - q ) (1 - q ) 160 2 q + ------------------------------------- 5 25 160 (1 - q) (1 - q ) (1 - q ) (1 - q ) 135 2 q + ------------------------------------- 5 50 135 (1 - q) (1 - q ) (1 - q ) (1 - q ) 85 2 q + ------------------------------------- 5 100 85 (1 - q) (1 - q ) (1 - q ) (1 - q ) 155 2 q + -------------------------------------- 10 25 155 (1 - q) (1 - q ) (1 - q ) (1 - q ) 130 2 q + -------------------------------------- 10 50 130 (1 - q) (1 - q ) (1 - q ) (1 - q ) 80 2 q + -------------------------------------- 10 100 80 (1 - q) (1 - q ) (1 - q ) (1 - q ) 115 2 q + -------------------------------------- 25 50 115 (1 - q) (1 - q ) (1 - q ) (1 - q ) 65 2 q + -------------------------------------- 25 100 65 (1 - q) (1 - q ) (1 - q ) (1 - q ) 40 2 q + -------------------------------------- 50 100 40 (1 - q) (1 - q ) (1 - q ) (1 - q ) 151 2 q + --------------------------------------- 5 10 25 151 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 126 2 q + --------------------------------------- 5 10 50 126 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 76 2 q + --------------------------------------- 5 10 100 76 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 111 2 q + --------------------------------------- 5 25 50 111 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 61 2 q + --------------------------------------- 5 25 100 61 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 36 2 q + --------------------------------------- 5 50 100 36 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 106 2 q + ---------------------------------------- 10 25 50 106 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 56 2 q + ---------------------------------------- 10 25 100 56 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 31 2 q + ---------------------------------------- 10 50 100 31 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 16 2 q + ---------------------------------------- 25 50 100 16 (1 - q ) (1 - q ) (1 - q ) (1 - q ) 150 q - ----------------------------------------------- 5 10 25 150 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 125 q - ----------------------------------------------- 5 10 50 125 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 75 q - ----------------------------------------------- 5 10 100 75 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 110 q - ----------------------------------------------- 5 25 50 110 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 60 q - ----------------------------------------------- 5 25 100 60 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 35 q - ----------------------------------------------- 5 50 100 35 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 105 q - ------------------------------------------------ 10 25 50 105 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 55 q - ------------------------------------------------ 10 25 100 55 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 30 q - ------------------------------------------------ 10 50 100 30 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 15 q - ------------------------------------------------ 25 50 100 15 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 101 q - ------------------------------------------------- 5 10 25 50 101 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 51 q - ------------------------------------------------- 5 10 25 100 51 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 26 q - ------------------------------------------------- 5 10 50 100 26 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 11 q - ------------------------------------------------- 5 25 50 100 11 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 6 q - ------------------------------------------------- 10 25 50 100 6 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) In Maple input format this is: -120*q^191/(1-q^191)+1/(1-q)/(1-q^5)*q^185/(1-q^35)/(1-q^150)+1/(1-q)/(1-q^5)*q ^185/(1-q^60)/(1-q^125)+1/(1-q)/(1-q^5)*q^185/(1-q^110)/(1-q^75)+1/(1-q)/(1-q^ 10)*q^180/(1-q^30)/(1-q^150)+1/(1-q)/(1-q^10)*q^180/(1-q^55)/(1-q^125)+1/(1-q)/ (1-q^10)*q^180/(1-q^105)/(1-q^75)+1/(1-q)/(1-q^25)*q^165/(1-q^15)/(1-q^150)+1/( 1-q)/(1-q^25)*q^165/(1-q^55)/(1-q^110)+1/(1-q)/(1-q^25)*q^165/(1-q^105)/(1-q^60 )+1/(1-q)/(1-q^50)*q^140/(1-q^15)/(1-q^125)+1/(1-q)/(1-q^50)*q^140/(1-q^30)/(1- q^110)+1/(1-q)/(1-q^50)*q^140/(1-q^105)/(1-q^35)+1/(1-q)/(1-q^100)*q^90/(1-q^15 )/(1-q^75)+1/(1-q)/(1-q^100)*q^90/(1-q^30)/(1-q^60)+1/(1-q)/(1-q^100)*q^90/(1-q ^55)/(1-q^35)+1/(1-q^5)/(1-q^10)*q^176/(1-q^26)/(1-q^150)+1/(1-q^5)/(1-q^10)*q^ 176/(1-q^51)/(1-q^125)+1/(1-q^5)/(1-q^10)*q^176/(1-q^101)/(1-q^75)+1/(1-q^5)/(1 -q^25)*q^161/(1-q^11)/(1-q^150)+1/(1-q^5)/(1-q^25)*q^161/(1-q^51)/(1-q^110)+1/( 1-q^5)/(1-q^25)*q^161/(1-q^101)/(1-q^60)+1/(1-q^5)/(1-q^50)*q^136/(1-q^11)/(1-q ^125)+1/(1-q^5)/(1-q^50)*q^136/(1-q^26)/(1-q^110)+1/(1-q^5)/(1-q^50)*q^136/(1-q ^101)/(1-q^35)+1/(1-q^5)/(1-q^100)*q^86/(1-q^11)/(1-q^75)+1/(1-q^5)/(1-q^100)*q ^86/(1-q^26)/(1-q^60)+1/(1-q^5)/(1-q^100)*q^86/(1-q^51)/(1-q^35)+1/(1-q^10)/(1- q^25)*q^156/(1-q^6)/(1-q^150)+1/(1-q^10)/(1-q^25)*q^156/(1-q^51)/(1-q^105)+1/(1 -q^10)/(1-q^25)*q^156/(1-q^101)/(1-q^55)+1/(1-q^10)/(1-q^50)*q^131/(1-q^6)/(1-q ^125)-2/(1-q^25)*q^166/(1-q^55)/(1-q^111)-2/(1-q^25)*q^166/(1-q^105)/(1-q^61)-2 /(1-q^25)*q^166/(1-q^101)/(1-q^65)-2/(1-q^25)*q^166/(1-q^15)/(1-q^151)-6/(1-q)/ (1-q^5)*q^185/(1-q^185)-6/(1-q)/(1-q^10)*q^180/(1-q^180)-6/(1-q)/(1-q^25)*q^165 /(1-q^165)-6/(1-q)/(1-q^50)*q^140/(1-q^140)-6/(1-q)/(1-q^100)*q^90/(1-q^90)-2/( 1-q)*q^190/(1-q^15)/(1-q^175)-2/(1-q)*q^190/(1-q^30)/(1-q^160)-2/(1-q)*q^190/(1 -q^55)/(1-q^135)-2/(1-q)*q^190/(1-q^105)/(1-q^85)-2/(1-q)*q^190/(1-q^35)/(1-q^ 155)-2/(1-q)*q^190/(1-q^60)/(1-q^130)-2/(1-q)*q^190/(1-q^110)/(1-q^80)-2/(1-q)* q^190/(1-q^75)/(1-q^115)-2/(1-q)*q^190/(1-q^125)/(1-q^65)-2/(1-q)*q^190/(1-q^ 150)/(1-q^40)-6/(1-q^5)/(1-q^10)*q^176/(1-q^176)-6/(1-q^5)/(1-q^25)*q^161/(1-q^ 161)-6/(1-q^5)/(1-q^50)*q^136/(1-q^136)-6/(1-q^5)/(1-q^100)*q^86/(1-q^86)-2/(1- q^5)*q^186/(1-q^11)/(1-q^175)-2/(1-q^5)*q^186/(1-q^26)/(1-q^160)-2/(1-q^5)*q^ 186/(1-q^51)/(1-q^135)-2/(1-q^5)*q^186/(1-q^101)/(1-q^85)-2/(1-q^5)*q^186/(1-q^ 35)/(1-q^151)-2/(1-q^5)*q^186/(1-q^60)/(1-q^126)-2/(1-q^5)*q^186/(1-q^110)/(1-q ^76)-2/(1-q^5)*q^186/(1-q^75)/(1-q^111)-2/(1-q^5)*q^186/(1-q^125)/(1-q^61)-2/(1 -q^5)*q^186/(1-q^150)/(1-q^36)-6/(1-q^10)/(1-q^25)*q^156/(1-q^156)-6/(1-q^10)/( 1-q^50)*q^131/(1-q^131)-6/(1-q^10)/(1-q^100)*q^81/(1-q^81)-2/(1-q^10)*q^181/(1- q^6)/(1-q^175)-2/(1-q^10)*q^181/(1-q^26)/(1-q^155)-2/(1-q^10)*q^181/(1-q^51)/(1 -q^130)-2/(1-q^10)*q^181/(1-q^101)/(1-q^80)-2/(1-q^10)*q^181/(1-q^30)/(1-q^151) -2/(1-q^10)*q^181/(1-q^55)/(1-q^126)-2/(1-q^10)*q^181/(1-q^105)/(1-q^76)-2/(1-q ^10)*q^181/(1-q^75)/(1-q^106)-2/(1-q^10)*q^181/(1-q^125)/(1-q^56)-2/(1-q^10)*q^ 181/(1-q^150)/(1-q^31)-6/(1-q^25)/(1-q^50)*q^116/(1-q^116)-6/(1-q^25)/(1-q^100) *q^66/(1-q^66)-2/(1-q^25)*q^166/(1-q^6)/(1-q^160)-2/(1-q^25)*q^166/(1-q^11)/(1- q^155)-2/(1-q^25)*q^166/(1-q^51)/(1-q^115)+1/(1-q^10)/(1-q^50)*q^131/(1-q^26)/( 1-q^105)+1/(1-q^10)/(1-q^50)*q^131/(1-q^101)/(1-q^30)+1/(1-q^10)/(1-q^100)*q^81 /(1-q^6)/(1-q^75)+1/(1-q^10)/(1-q^100)*q^81/(1-q^26)/(1-q^55)+1/(1-q^10)/(1-q^ 100)*q^81/(1-q^51)/(1-q^30)+1/(1-q^25)/(1-q^50)*q^116/(1-q^6)/(1-q^110)+1/(1-q^ 25)/(1-q^50)*q^116/(1-q^11)/(1-q^105)+1/(1-q^25)/(1-q^50)*q^116/(1-q^101)/(1-q^ 15)+1/(1-q^25)/(1-q^100)*q^66/(1-q^6)/(1-q^60)+1/(1-q^25)/(1-q^100)*q^66/(1-q^ 11)/(1-q^55)+1/(1-q^25)/(1-q^100)*q^66/(1-q^51)/(1-q^15)+1/(1-q^50)/(1-q^100)*q ^41/(1-q^6)/(1-q^35)+1/(1-q^50)/(1-q^100)*q^41/(1-q^11)/(1-q^30)+1/(1-q^50)/(1- q^100)*q^41/(1-q^26)/(1-q^15)+1/(1-q)/(1-q^5)/(1-q^10)/(1-q^25)/(1-q^50)/(1-q^ 100)+24/(1-q)*q^190/(1-q^190)+24/(1-q^5)*q^186/(1-q^186)+24/(1-q^10)*q^181/(1-q ^181)+24/(1-q^25)*q^166/(1-q^166)+24/(1-q^50)*q^141/(1-q^141)+24/(1-q^100)*q^91 /(1-q^91)+6*q^191/(1-q^6)/(1-q^185)+6*q^191/(1-q^11)/(1-q^180)+6*q^191/(1-q^26) /(1-q^165)+6*q^191/(1-q^51)/(1-q^140)+6*q^191/(1-q^101)/(1-q^90)+6*q^191/(1-q^ 15)/(1-q^176)+6*q^191/(1-q^30)/(1-q^161)+6*q^191/(1-q^55)/(1-q^136)+6*q^191/(1- q^105)/(1-q^86)+6*q^191/(1-q^35)/(1-q^156)+6*q^191/(1-q^60)/(1-q^131)+6*q^191/( 1-q^110)/(1-q^81)+6*q^191/(1-q^75)/(1-q^116)+6*q^191/(1-q^125)/(1-q^66)+6*q^191 /(1-q^150)/(1-q^41)+4*q^191/(1-q^16)/(1-q^175)+4*q^191/(1-q^31)/(1-q^160)+4*q^ 191/(1-q^56)/(1-q^135)+4*q^191/(1-q^106)/(1-q^85)+4*q^191/(1-q^36)/(1-q^155)+4* q^191/(1-q^61)/(1-q^130)+4*q^191/(1-q^111)/(1-q^80)+4*q^191/(1-q^76)/(1-q^115)+ 4*q^191/(1-q^126)/(1-q^65)+4*q^191/(1-q^151)/(1-q^40)-2/(1-q^25)*q^166/(1-q^60) /(1-q^106)-2/(1-q^25)*q^166/(1-q^110)/(1-q^56)-2/(1-q^25)*q^166/(1-q^150)/(1-q^ 16)-6/(1-q^50)/(1-q^100)*q^41/(1-q^41)-2/(1-q^50)*q^141/(1-q^6)/(1-q^135)-2/(1- q^50)*q^141/(1-q^11)/(1-q^130)-2/(1-q^50)*q^141/(1-q^26)/(1-q^115)-2/(1-q^50)*q ^141/(1-q^101)/(1-q^40)-2/(1-q^50)*q^141/(1-q^15)/(1-q^126)-2/(1-q^50)*q^141/(1 -q^30)/(1-q^111)-2/(1-q^50)*q^141/(1-q^105)/(1-q^36)-2/(1-q^50)*q^141/(1-q^35)/ (1-q^106)-2/(1-q^50)*q^141/(1-q^110)/(1-q^31)-2/(1-q^50)*q^141/(1-q^125)/(1-q^ 16)-2/(1-q^100)*q^91/(1-q^6)/(1-q^85)-2/(1-q^100)*q^91/(1-q^11)/(1-q^80)-2/(1-q ^100)*q^91/(1-q^26)/(1-q^65)-2/(1-q^100)*q^91/(1-q^51)/(1-q^40)-2/(1-q^100)*q^ 91/(1-q^15)/(1-q^76)-2/(1-q^100)*q^91/(1-q^30)/(1-q^61)-2/(1-q^100)*q^91/(1-q^ 55)/(1-q^36)-2/(1-q^100)*q^91/(1-q^35)/(1-q^56)-2/(1-q^100)*q^91/(1-q^60)/(1-q^ 31)-2/(1-q^100)*q^91/(1-q^75)/(1-q^16)-q^191/(1-q^6)/(1-q^35)/(1-q^150)-q^191/( 1-q^6)/(1-q^60)/(1-q^125)-q^191/(1-q^6)/(1-q^110)/(1-q^75)-q^191/(1-q^11)/(1-q^ 30)/(1-q^150)-q^191/(1-q^11)/(1-q^55)/(1-q^125)-q^191/(1-q^11)/(1-q^105)/(1-q^ 75)-q^191/(1-q^26)/(1-q^15)/(1-q^150)-q^191/(1-q^26)/(1-q^55)/(1-q^110)-q^191/( 1-q^26)/(1-q^105)/(1-q^60)-q^191/(1-q^51)/(1-q^15)/(1-q^125)-q^191/(1-q^51)/(1- q^30)/(1-q^110)-q^191/(1-q^51)/(1-q^105)/(1-q^35)-q^191/(1-q^101)/(1-q^15)/(1-q ^75)-q^191/(1-q^101)/(1-q^30)/(1-q^60)-q^191/(1-q^101)/(1-q^55)/(1-q^35)+2/(1-q )/(1-q^5)/(1-q^10)*q^175/(1-q^175)+2/(1-q)/(1-q^5)/(1-q^25)*q^160/(1-q^160)+2/( 1-q)/(1-q^5)/(1-q^50)*q^135/(1-q^135)+2/(1-q)/(1-q^5)/(1-q^100)*q^85/(1-q^85)+2 /(1-q)/(1-q^10)/(1-q^25)*q^155/(1-q^155)+2/(1-q)/(1-q^10)/(1-q^50)*q^130/(1-q^ 130)+2/(1-q)/(1-q^10)/(1-q^100)*q^80/(1-q^80)+2/(1-q)/(1-q^25)/(1-q^50)*q^115/( 1-q^115)+2/(1-q)/(1-q^25)/(1-q^100)*q^65/(1-q^65)+2/(1-q)/(1-q^50)/(1-q^100)*q^ 40/(1-q^40)+2/(1-q^5)/(1-q^10)/(1-q^25)*q^151/(1-q^151)+2/(1-q^5)/(1-q^10)/(1-q ^50)*q^126/(1-q^126)+2/(1-q^5)/(1-q^10)/(1-q^100)*q^76/(1-q^76)+2/(1-q^5)/(1-q^ 25)/(1-q^50)*q^111/(1-q^111)+2/(1-q^5)/(1-q^25)/(1-q^100)*q^61/(1-q^61)+2/(1-q^ 5)/(1-q^50)/(1-q^100)*q^36/(1-q^36)+2/(1-q^10)/(1-q^25)/(1-q^50)*q^106/(1-q^106 )+2/(1-q^10)/(1-q^25)/(1-q^100)*q^56/(1-q^56)+2/(1-q^10)/(1-q^50)/(1-q^100)*q^ 31/(1-q^31)+2/(1-q^25)/(1-q^50)/(1-q^100)*q^16/(1-q^16)-1/(1-q)/(1-q^5)/(1-q^10 )/(1-q^25)*q^150/(1-q^150)-1/(1-q)/(1-q^5)/(1-q^10)/(1-q^50)*q^125/(1-q^125)-1/ (1-q)/(1-q^5)/(1-q^10)/(1-q^100)*q^75/(1-q^75)-1/(1-q)/(1-q^5)/(1-q^25)/(1-q^50 )*q^110/(1-q^110)-1/(1-q)/(1-q^5)/(1-q^25)/(1-q^100)*q^60/(1-q^60)-1/(1-q)/(1-q ^5)/(1-q^50)/(1-q^100)*q^35/(1-q^35)-1/(1-q)/(1-q^10)/(1-q^25)/(1-q^50)*q^105/( 1-q^105)-1/(1-q)/(1-q^10)/(1-q^25)/(1-q^100)*q^55/(1-q^55)-1/(1-q)/(1-q^10)/(1- q^50)/(1-q^100)*q^30/(1-q^30)-1/(1-q)/(1-q^25)/(1-q^50)/(1-q^100)*q^15/(1-q^15) -1/(1-q^5)/(1-q^10)/(1-q^25)/(1-q^50)*q^101/(1-q^101)-1/(1-q^5)/(1-q^10)/(1-q^ 25)/(1-q^100)*q^51/(1-q^51)-1/(1-q^5)/(1-q^10)/(1-q^50)/(1-q^100)*q^26/(1-q^26) -1/(1-q^5)/(1-q^25)/(1-q^50)/(1-q^100)*q^11/(1-q^11)-1/(1-q^10)/(1-q^25)/(1-q^ 50)/(1-q^100)*q^6/(1-q^6) For the sake of Sloane, the first, 1000, terms are [1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 3, 4, 4, 5, 5, 5, 4, 5, 8, 7, 6, 8, 7, 12, 9, 11, 11, 12, 14, 14, 15, 14, 14, 20, 16, 18, 18, 20, 25, 24, 23, 24, 24, 32, 27, 30, 31, 30, 40, 38, 37, 40, 38, 48, 44, 48, 46, 49, 57, 54, 54, 58, 59, 72, 63, 68, 67, 70, 83, 80, 79, 84, 81, 100, 90, 96, 95, 98, 117, 110, 112, 115, 115, 139, 125, 132, 132, 141, 157, 148, 150, 157, 155, 185, 171, 176, 177, 181, 213, 201, 201, 209, 207, 245, 225, 238, 236, 240, 273, 261, 268, 274, 274, 315, 292, 305, 303, 313, 354, 337, 340, 352, 347, 407, 377, 390, 393, 402, 452, 432, 437, 447, 447, 511, 475, 496, 495, 508, 562, 541, 544, 561, 560, 634, 598, 613, 616, 626, 699, 668, 675, 698, 689, 782, 738, 760, 757, 776, 854, 826, 838, 851, 852, 957, 896, 928, 928, 946, 1039, 1007, 1011, 1039, 1032, 1160, 1096, 1126, 1128, 1149, 1260, 1210, 1224, 1255, 1246, 1394, 1317, 1354, 1353, 1389, 1497, 1451, 1463, 1493, 1496, 1660, 1575, 1610, 1616, 1639, 1794, 1738, 1742, 1786, 1778, 1969, 1872, 1921, 1920, 1955, 2122, 2046, 2072, 2111, 2109, 2323, 2216, 2267, 2262, 2306, 2493, 2422, 2432, 2479, 2477, 2733, 2600, 2658, 2675, 2705, 2921, 2839, 2853, 2907, 2905, 3186, 3037, 3110, 3112, 3169, 3407, 3301, 3326, 3383, 3394, 3701, 3537, 3609, 3621, 3670, 3952, 3838, 3858, 3934, 3918, 4287, 4099, 4188, 4195, 4247, 4563, 4435, 4473, 4534, 4544, 4938, 4729, 4829, 4834, 4903, 5247, 5111, 5135, 5221, 5217, 5672, 5451, 5543, 5559, 5624, 6022, 5862, 5898, 5982, 5977, 6485, 6229, 6348, 6364, 6448, 6866, 6685, 6742, 6830, 6846, 7382, 7109, 7228, 7257, 7325, 7832, 7631, 7662, 7785, 7778, 8394, 8091, 8234, 8254, 8337, 8876, 8652, 8723, 8824, 8835, 9499, 9164, 9322, 9343, 9450, 10034, 9797, 9855, 9975, 9990, 10734, 10361, 10521, 10568, 10664, 11326, 11051, 11116, 11260, 11261, 12083, 11677, 11862, 11895, 12019, 12726, 12431, 12508, 12649, 12679, 13556, 13118, 13308, 13361, 13476, 14272, 13951, 14033, 14206, 14207, 15185, 14711, 14917, 14969, 15096, 15977, 15605, 15710, 15876, 15907, 16958, 16439, 16680, 16712, 16862, 17810, 17433, 17531, 17712, 17730, 18900, 18349, 18576, 18649, 18796, 19835, 19410, 19522, 19726, 19746, 21018, 20393, 20663, 20738, 20917, 22025, 21562, 21693, 21896, 21953, 23301, 22656, 22926, 23003, 23183, 24435, 23934, 24045, 24309, 24314, 25808, 25103, 25409, 25505, 25691, 27025, 26473, 26644, 26878, 26928, 28520, 27750, 28087, 28169, 28393, 29850, 29261, 29393, 29671, 29727, 31478, 30647, 30990, 31097, 31326, 32901, 32272, 32429, 32722, 32761, 34666, 33764, 34142, 34244, 34527, 36198, 35503, 35690, 35997, 36091, 38096, 37131, 37529, 37657, 37918, 39787, 39037, 39206, 39573, 39622, 41818, 40795, 41224, 41354, 41632, 43619, 42820, 43041, 43397, 43468, 45836, 44714, 45180, 45306, 45637, 47751, 46906, 47121, 47504, 47599, 50147, 48956, 49416, 49599, 49920, 52257, 51320, 51523, 51963, 52049, 54789, 53501, 54024, 54188, 54575, 57022, 56017, 56279, 56715, 56848, 59765, 58403, 58921, 59137, 59504, 62187, 61123, 61366, 61873, 61959, 65112, 63656, 64228, 64433, 64834, 67723, 66566, 66855, 67353, 67487, 70828, 69277, 69918, 70094, 70562, 73624, 72403, 72685, 73236, 73354, 76995, 75344, 75956, 76214, 76689, 79948, 78661, 78979, 79541, 79679, 83551, 81766, 82466, 82716, 83233, 86726, 85323, 85663, 86266, 86462, 90527, 88660, 89398, 89652, 90188, 93959, 92460, 92809, 93491, 93606, 98020, 96034, 96799, 97102, 97634, 101651, 100058, 100484, 101141, 101343, 105991, 103848, 104703, 104978, 105586, 109859, 108185, 108581, 109313, 109514, 114493, 112264, 113102, 113430, 114049, 118636, 116816, 117255, 118050, 118221, 123550, 121143, 122062, 122409, 123123, 127894, 126007, 126487, 127263, 127544, 133131, 130609, 131564, 131941, 132628, 137851, 135831, 136267, 137180, 137384, 143348, 140712, 141734, 142111, 142833, 148343, 146174, 146743, 147633, 147879, 154195, 151390, 152488, 152862, 153680, 159468, 157229, 157788, 158700, 158985, 165735, 162754, 163879, 164332, 165131, 171340, 168931, 169507, 170507, 170793, 177918, 174778, 175992, 176445, 177344, 183861, 181304, 181952, 182974, 183323, 190820, 187527, 188755, 189266, 190192, 197129, 194449, 195122, 196238, 196530, 204507, 201019, 202344, 202871, 203782, 211193, 208357, 209073, 210202, 210581, 218936, 215298, 216716, 217188, 218243, 226005, 223039, 223788, 224973, 225336, 234250, 230418, 231836, 232440, 233480, 241706, 238576, 239371, 240622, 241002, 250411, 246320, 247873, 248479, 249614, 258264, 254961, 255805, 257094, 257586, 267387, 263184, 264764, 265392, 266562, 275757, 272279, 273136, 274573, 274969, 285392, 280947, 282615, 283313, 284501, 294176, 290495, 291481, 292889, 293395, 304298, 299643, 301460, 302097, 303406, 313599, 309765, 310710, 312225, 312740, 324282, 319421, 321238, 321987, 323329, 334043, 330003, 331054, 332620, 333134, 345307, 340160, 342101, 342885, 344329, 355560, 351304, 352412, 354018, 354690, 367359, 361995, 364001, 364847, 366306, 378226, 373805, 374889, 376662, 377223, 390599, 385019, 387129, 387991, 389481, 402014, 397321, 398596, 400346, 400980, 415040, 409172, 411379, 412243, 413880, 426964, 422142, 423378, 425233, 425910, 440681, 434585, 436821, 437796, 439411, 453274, 448194, 449495, 451430, 452117, 467619, 461185, 463597, 464578, 466379, 480768, 475424, 476867, 478844, 479649, 495826, 489146, 491601, 492685, 494483, 509689, 504142, 505530, 507713, 508455, 525445, 518485, 521089, 522203, 524017, 539998, 534150, 535709, 537896, 538734, 556449, 549228, 551976, 553037, 555037, 571694, 565625, 567214, 569490, 570336, 589045, 581469, 584228, 585488, 587489, 604936, 598635, 600275, 602629, 603534, 623050, 615135, 618089, 619333, 621489, 639735, 633118, 634848, 637298, 638330, 658599, 650444, 653456, 654781, 656959, 676130, 669223, 671031, 673687, 674562, 695895, 687366, 690513, 691925, 694146, 714185, 707004, 708923, 711565, 712655, 734815, 725916, 729261, 730627, 733037, 753951, 746508, 748449, 751229, 752293, 775558, 766355, 769700, 771239, 773647, 795553, 787759, 789833, 792728, 793791, 818111, 808480, 811989, 813580, 816209, 838910, 830822, 832989, 835932, 837232, 862431, 852466, 856079, 857765, 860375, 884291, 875892, 878013, 881231, 882398, 908766, 898431, 902227, 903962, 906642, 931526, 922744, 925128, 928352, 929633, 957065, 946359, 950297, 952021, 954933, 980792, 971731, 974112, 977427, 978788, 1007518, 996351, 1000387, 1002273, 1005167, 1032269, 1022829, 1025281, 1028745, 1030138, 1060017, 1048427, 1052675, 1054587, 1057705, 1085787, 1075959, 1078603, 1082123, 1083701, 1114680, 1102722, 1107018, 1109101, 1112246, 1141598, 1131437, 1134070, 1137878, 1139338, 1171676, 1159273, 1163789, 1165895, 1169074, 1199759, 1189157, 1191971, 1195819, 1197455, 1230972, 1218140, 1222840, 1224935, 1228389, 1260149, 1249225, 1252122, 1256124, 1257718, 1292779, 1279495, 1284210, 1286531, 1290013, 1323126, 1311820, 1314784, 1318875, 1320593, 1357037, 1343168, 1348140, 1350502, 1354215, 1388561, 1376828, 1379918, 1384099, 1386043, 1423698, 1409518, 1414564, 1417040, 1420776, 1456633] For the sake of comparison the generating function for the number of ways of making change without restiction is, of course, the much simpler 1 --------------------------------------------------------- 5 10 25 50 100 (1 - q) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) For the sake of Sloane, the first, 1000, terms are [1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 13, 13, 13, 13, 13, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 31, 31, 31, 31, 31, 39, 39, 39, 39, 39, 50, 50, 50, 50, 50, 62, 62, 62, 62, 62, 77, 77, 77, 77, 77, 93, 93, 93, 93, 93, 112, 112, 112, 112, 112, 134, 134, 134, 134, 134, 159, 159, 159, 159, 159, 187, 187, 187, 187, 187, 218, 218, 218, 218, 218, 252, 252, 252, 252, 252, 293, 293, 293, 293, 293, 337, 337, 337, 337, 337, 388, 388, 388, 388, 388, 442, 442, 442, 442, 442, 503, 503, 503, 503, 503, 571, 571, 571, 571, 571, 646, 646, 646, 646, 646, 728, 728, 728, 728, 728, 817, 817, 817, 817, 817, 913, 913, 913, 913, 913, 1022, 1022, 1022, 1022, 1022, 1138, 1138, 1138, 1138, 1138, 1267, 1267, 1267, 1267, 1267, 1403, 1403, 1403, 1403, 1403, 1552, 1552, 1552, 1552, 1552, 1714, 1714, 1714, 1714, 1714, 1889, 1889, 1889, 1889, 1889, 2077, 2077, 2077, 2077, 2077, 2278, 2278, 2278, 2278, 2278, 2492, 2492, 2492, 2492, 2492, 2728, 2728, 2728, 2728, 2728, 2977, 2977, 2977, 2977, 2977, 3248, 3248, 3248, 3248, 3248, 3532, 3532, 3532, 3532, 3532, 3838, 3838, 3838, 3838, 3838, 4166, 4166, 4166, 4166, 4166, 4516, 4516, 4516, 4516, 4516, 4888, 4888, 4888, 4888, 4888, 5282, 5282, 5282, 5282, 5282, 5698, 5698, 5698, 5698, 5698, 6148, 6148, 6148, 6148, 6148, 6620, 6620, 6620, 6620, 6620, 7126, 7126, 7126, 7126, 7126, 7654, 7654, 7654, 7654, 7654, 8216, 8216, 8216, 8216, 8216, 8812, 8812, 8812, 8812, 8812, 9442, 9442, 9442, 9442, 9442, 10106, 10106, 10106, 10106, 10106, 10804, 10804, 10804, 10804, 10804, 11536, 11536, 11536, 11536, 11536, 12318, 12318, 12318, 12318, 12318, 13134, 13134, 13134, 13134, 13134, 14000, 14000, 14000, 14000, 14000, 14900, 14900, 14900, 14900, 14900, 15850, 15850, 15850, 15850, 15850, 16850, 16850, 16850, 16850, 16850, 17900, 17900, 17900, 17900, 17900, 19000, 19000, 19000, 19000, 19000, 20150, 20150, 20150, 20150, 20150, 21350, 21350, 21350, 21350, 21350, 22620, 22620, 22620, 22620, 22620, 23940, 23940, 23940, 23940, 23940, 25330, 25330, 25330, 25330, 25330, 26770, 26770, 26770, 26770, 26770, 28280, 28280, 28280, 28280, 28280, 29860, 29860, 29860, 29860, 29860, 31510, 31510, 31510, 31510, 31510, 33230, 33230, 33230, 33230, 33230, 35020, 35020, 35020, 35020, 35020, 36880, 36880, 36880, 36880, 36880, 38835, 38835, 38835, 38835, 38835, 40860, 40860, 40860, 40860, 40860, 42980, 42980, 42980, 42980, 42980, 45170, 45170, 45170, 45170, 45170, 47455, 47455, 47455, 47455, 47455, 49835, 49835, 49835, 49835, 49835, 52310, 52310, 52310, 52310, 52310, 54880, 54880, 54880, 54880, 54880, 57545, 57545, 57545, 57545, 57545, 60305, 60305, 60305, 60305, 60305, 63190, 63190, 63190, 63190, 63190, 66170, 66170, 66170, 66170, 66170, 69275, 69275, 69275, 69275, 69275, 72475, 72475, 72475, 72475, 72475, 75800, 75800, 75800, 75800, 75800, 79250, 79250, 79250, 79250, 79250, 82825, 82825, 82825, 82825, 82825, 86525, 86525, 86525, 86525, 86525, 90350, 90350, 90350, 90350, 90350, 94300, 94300, 94300, 94300, 94300, 98411, 98411, 98411, 98411, 98411, 102647, 102647, 102647, 102647, 102647, 107044, 107044, 107044, 107044, 107044, 111566, 111566, 111566, 111566, 111566, 116249, 116249, 116249, 116249, 116249, 121093, 121093, 121093, 121093, 121093, 126098, 126098, 126098, 126098, 126098, 131264, 131264, 131264, 131264, 131264, 136591, 136591, 136591, 136591, 136591, 142079, 142079, 142079, 142079, 142079, 147770, 147770, 147770, 147770, 147770, 153622, 153622, 153622, 153622, 153622, 159677, 159677, 159677, 159677, 159677, 165893, 165893, 165893, 165893, 165893, 172312, 172312, 172312, 172312, 172312, 178934, 178934, 178934, 178934, 178934, 185759, 185759, 185759, 185759, 185759, 192787, 192787, 192787, 192787, 192787, 200018, 200018, 200018, 200018, 200018, 207452, 207452, 207452, 207452, 207452, 215138, 215138, 215138, 215138, 215138, 223027, 223027, 223027, 223027, 223027, 231168, 231168, 231168, 231168, 231168, 239512, 239512, 239512, 239512, 239512, 248108, 248108, 248108, 248108, 248108, 256956, 256956, 256956, 256956, 256956, 266056, 266056, 266056, 266056, 266056, 275408, 275408, 275408, 275408, 275408, 285012, 285012, 285012, 285012, 285012, 294868, 294868, 294868, 294868, 294868, 305032, 305032, 305032, 305032, 305032, 315448, 315448, 315448, 315448, 315448, 326172, 326172, 326172, 326172, 326172, 337148, 337148, 337148, 337148, 337148, 348432, 348432, 348432, 348432, 348432, 360024, 360024, 360024, 360024, 360024, 371924, 371924, 371924, 371924, 371924, 384132, 384132, 384132, 384132, 384132, 396648, 396648, 396648, 396648, 396648, 409472, 409472, 409472, 409472, 409472, 422668, 422668, 422668, 422668, 422668, 436172, 436172, 436172, 436172, 436172, 450048, 450048, 450048, 450048, 450048, 464232, 464232, 464232, 464232, 464232, 478788, 478788, 478788, 478788, 478788, 493716, 493716, 493716, 493716, 493716, 509016, 509016, 509016, 509016, 509016, 524688, 524688, 524688, 524688, 524688, 540732, 540732, 540732, 540732, 540732, 557148, 557148, 557148, 557148, 557148, 574008, 574008, 574008, 574008, 574008, 591240, 591240, 591240, 591240, 591240, 608916, 608916, 608916, 608916, 608916, 626964, 626964, 626964, 626964, 626964, 645456, 645456, 645456, 645456, 645456, 664392, 664392, 664392, 664392, 664392, 683772, 683772, 683772, 683772, 683772, 703596, 703596, 703596, 703596, 703596, 723864, 723864, 723864, 723864, 723864, 744576, 744576, 744576, 744576, 744576, 765813, 765813, 765813, 765813, 765813, 787494, 787494, 787494, 787494, 787494, 809700, 809700, 809700, 809700, 809700, 832350, 832350, 832350, 832350, 832350, 855525, 855525, 855525, 855525, 855525, 879225, 879225, 879225, 879225, 879225, 903450, 903450, 903450, 903450, 903450, 928200, 928200, 928200, 928200, 928200, 953475, 953475, 953475, 953475, 953475, 979275, 979275, 979275, 979275, 979275, 1005690, 1005690, 1005690, 1005690, 1005690, 1032630, 1032630, 1032630, 1032630, 1032630, 1060185, 1060185, 1060185, 1060185, 1060185, 1088265, 1088265, 1088265, 1088265, 1088265, 1116960, 1116960, 1116960, 1116960, 1116960, 1146270, 1146270, 1146270, 1146270, 1146270, 1176195, 1176195, 1176195, 1176195, 1176195, 1206735, 1206735, 1206735, 1206735, 1206735, 1237890, 1237890, 1237890, 1237890, 1237890, 1269660, 1269660, 1269660, 1269660, 1269660, 1302145, 1302145, 1302145, 1302145, 1302145, 1335245, 1335245, 1335245, 1335245, 1335245, 1369060, 1369060, 1369060, 1369060, 1369060, 1403490, 1403490, 1403490, 1403490, 1403490, 1438635, 1438635, 1438635, 1438635, 1438635, 1474495, 1474495, 1474495, 1474495, 1474495, 1511070, 1511070, 1511070, 1511070, 1511070, 1548360, 1548360, 1548360, 1548360, 1548360, 1586365, 1586365, 1586365, 1586365, 1586365, 1625085, 1625085, 1625085, 1625085, 1625085, 1664630, 1664630, 1664630, 1664630, 1664630, 1704890, 1704890, 1704890, 1704890, 1704890, 1745975, 1745975, 1745975, 1745975, 1745975, 1787775, 1787775, 1787775, 1787775, 1787775, 1830400, 1830400, 1830400, 1830400, 1830400, 1873850, 1873850, 1873850, 1873850, 1873850, 1918125, 1918125, 1918125, 1918125, 1918125, 1963225, 1963225, 1963225, 1963225, 1963225, 2009150, 2009150, 2009150, 2009150, 2009150, 2055900, 2055900, 2055900, 2055900, 2055900, 2103596] The sequence of ratios of good/all is [1., 1., 1., 1., 1., 0.5000000000, 1., 1., 1., 1., 0.7500000000, 0.7500000000, 1., 1., 0.8333333333, 0.8333333333, 0.8333333333, 0.6666666667, 0.8333333333, 0.8888888889, 0.7777777778, 0.6666666667, 0.8888888889, 0.7777777778, 0.9230769231, 0.6923076923, 0.8461538462, 0.8461538462, 0.9230769231, 0.7777777778, 0.7777777778, 0.8333333333, 0.7777777778, 0.7777777778, 0.8333333333, 0.6666666667, 0.7500000000, 0.7500000000, 0.8333333333, 0.8064516129, 0.7741935484, 0.7419354839, 0.7741935484, 0.7741935484, 0.8205128205, 0.6923076923, 0.7692307692, 0.7948717949, 0.7692307692, 0.8000000000, 0.7600000000, 0.7400000000, 0.8000000000, 0.7600000000, 0.7741935484, 0.7096774194, 0.7741935484, 0.7419354839, 0.7903225806, 0.7402597403, 0.7012987013, 0.7012987013, 0.7532467532, 0.7662337662, 0.7741935484, 0.6774193548, 0.7311827957, 0.7204301075, 0.7526881720, 0.7410714286, 0.7142857143, 0.7053571429, 0.7500000000, 0.7232142857, 0.7462686567, 0.6716417910, 0.7164179104, 0.7089552239, 0.7313432836, 0.7358490566, 0.6918238994, 0.7044025157, 0.7232704403, 0.7232704403, 0.7433155080, 0.6684491979, 0.7058823529, 0.7058823529, 0.7540106952, 0.7201834862, 0.6788990826, 0.6880733945, 0.7201834862, 0.7110091743, 0.7341269841, 0.6785714286, 0.6984126984, 0.7023809524, 0.7182539683, 0.7269624573, 0.6860068259, 0.6860068259, 0.7133105802, 0.7064846416, 0.7270029674, 0.6676557864, 0.7062314540, 0.7002967359, 0.7121661721, 0.7036082474, 0.6726804124, 0.6907216495, 0.7061855670, 0.7061855670, 0.7126696833, 0.6606334842, 0.6900452489, 0.6855203620, 0.7081447964, 0.7037773360, 0.6699801193, 0.6759443340, 0.6998011928, 0.6898608350, 0.7127845884, 0.6602451839, 0.6830122592, 0.6882661996, 0.7040280210, 0.6996904025, 0.6687306502, 0.6764705882, 0.6919504644, 0.6919504644, 0.7019230769, 0.6524725275, 0.6813186813, 0.6799450549, 0.6978021978, 0.6878824969, 0.6621787026, 0.6658506732, 0.6866585067, 0.6854345165, 0.6944140197, 0.6549835706, 0.6714129244, 0.6746987952, 0.6856516977, 0.6839530333, 0.6536203522, 0.6604696673, 0.6829745597, 0.6741682975, 0.6871704745, 0.6485061511, 0.6678383128, 0.6652021090, 0.6818980668, 0.6740331492, 0.6519337017, 0.6614048934, 0.6716653512, 0.6724546172, 0.6821097648, 0.6386315039, 0.6614397719, 0.6614397719, 0.6742694227, 0.6694587629, 0.6488402062, 0.6514175258, 0.6694587629, 0.6649484536, 0.6767794632, 0.6394399067, 0.6569428238, 0.6581096849, 0.6703617270, 0.6670195871, 0.6405505558, 0.6479618846, 0.6643726840, 0.6596082583, 0.6711603274, 0.6340876264, 0.6519017814, 0.6514203178, 0.6687530091, 0.6571553995, 0.6369622476, 0.6422300263, 0.6553994732, 0.6567164179, 0.6661316212, 0.6320224719, 0.6460674157, 0.6484751204, 0.6577046549, 0.6576246334, 0.6370967742, 0.6385630499, 0.6546920821, 0.6517595308, 0.6614040981, 0.6288209607, 0.6452804837, 0.6449445751, 0.6567013772, 0.6533251232, 0.6299261084, 0.6379310345, 0.6499384236, 0.6493226601, 0.6577010193, 0.6274065685, 0.6418459796, 0.6404303511, 0.6528878822, 0.6495570610, 0.6310578426, 0.6336633663, 0.6459093278, 0.6453882230, 0.6560249640, 0.6240998560, 0.6380220835, 0.6421027364, 0.6493038886, 0.6468113375, 0.6286536758, 0.6317537644, 0.6437112489, 0.6432683791, 0.6518003273, 0.6213175123, 0.6362520458, 0.6366612111, 0.6483224223, 0.6450208254, 0.6249526694, 0.6296857251, 0.6404770920, 0.6425596365, 0.6495261495, 0.6207441207, 0.6333801334, 0.6354861355, 0.6440856441, 0.6428106701, 0.6242680547, 0.6275211451, 0.6398828887, 0.6372804164, 0.6475830816, 0.6191842900, 0.6326283988, 0.6336858006, 0.6415407855, 0.6403311816, 0.6223687903, 0.6277013752, 0.6362615773, 0.6376648891, 0.6451528612, 0.6178468774, 0.6309119415, 0.6315651947, 0.6405800888, 0.6386319377, 0.6220788705, 0.6250000000, 0.6354673807, 0.6349805258, 0.6436677258, 0.6185882887, 0.6290285974, 0.6308443032, 0.6382206083, 0.6377886041, 0.6208430417, 0.6246557933, 0.6335522135, 0.6330226647, 0.6416980012, 0.6163665149, 0.6281416980, 0.6297249159, 0.6380368098, 0.6355053684, 0.6187523140, 0.6240281377, 0.6321732692, 0.6336542021, 0.6399098474, 0.6162447989, 0.6265603329, 0.6290742025, 0.6349687933, 0.6358175028, 0.6194999188, 0.6220165611, 0.6320019484, 0.6314336743, 0.6391046140, 0.6160347190, 0.6269224912, 0.6284452566, 0.6347647328, 0.6340000000, 0.6180000000, 0.6230714286, 0.6302857143, 0.6310714286, 0.6375167785, 0.6150335570, 0.6256375839, 0.6270469799, 0.6342281879, 0.6330599369, 0.6181072555, 0.6217665615, 0.6293375394, 0.6302839117, 0.6370326409, 0.6148961424, 0.6243916914, 0.6271810089, 0.6328783383, 0.6327374302, 0.6173743017, 0.6210055866, 0.6290502793, 0.6291061453, 0.6359473684, 0.6145789474, 0.6243157895, 0.6260526316, 0.6325789474, 0.6315632754, 0.6169230769, 0.6207444169, 0.6277419355, 0.6292307692, 0.6349414520, 0.6144262295, 0.6233255269, 0.6258079625, 0.6311943794, 0.6309460654, 0.6167550840, 0.6203801945, 0.6280282935, 0.6280725022, 0.6342940685, 0.6144945698, 0.6230994152, 0.6252715121, 0.6305764411, 0.6307540466, 0.6160679037, 0.6202131859, 0.6267666798, 0.6279905251, 0.6334703026, 0.6140829287, 0.6230855435, 0.6242809115, 0.6298841987, 0.6297736917, 0.6164427157, 0.6199080622, 0.6263083451, 0.6269448373, 0.6329537843, 0.6145010047, 0.6221031480, 0.6245478902, 0.6294708640, 0.6294827039, 0.6159949222, 0.6195493494, 0.6260234846, 0.6266582037, 0.6325007523, 0.6136924466, 0.6218176347, 0.6240746314, 0.6294613301, 0.6289263278, 0.6157053112, 0.6194460308, 0.6252427184, 0.6268703598, 0.6318058568, 0.6143167028, 0.6216377440, 0.6237255965, 0.6286062907, 0.6292004635, 0.6162997296, 0.6191579761, 0.6259559676, 0.6260847174, 0.6316201664, 0.6143661282, 0.6218551150, 0.6242046011, 0.6287567303, 0.6287808283, 0.6159376454, 0.6199162401, 0.6253606329, 0.6265239646, 0.6313925172, 0.6143458047, 0.6218065087, 0.6236218729, 0.6285809165, 0.6290169634, 0.6166052049, 0.6193867875, 0.6252449689, 0.6264250342, 0.6316444266, 0.6149693990, 0.6218521120, 0.6239991974, 0.6285943614, 0.6289619576, 0.6169374881, 0.6199388262, 0.6255400497, 0.6262856050, 0.6316690962, 0.6152332362, 0.6221209913, 0.6239795918, 0.6291362974, 0.6290381441, 0.6169606395, 0.6202102702, 0.6255452255, 0.6271787297, 0.6317220794, 0.6157200895, 0.6223198740, 0.6244424177, 0.6287704170, 0.6296407659, 0.6177717993, 0.6204462731, 0.6262541541, 0.6270295933, 0.6319782379, 0.6165180595, 0.6230013601, 0.6249659967, 0.6291672964, 0.6296499459, 0.6181162035, 0.6213063876, 0.6264453266, 0.6274702274, 0.6324387720, 0.6169575716, 0.6233873750, 0.6251259055, 0.6296929976, 0.6299604222, 0.6188126649, 0.6216490765, 0.6267018470, 0.6279551451, 0.6327697161, 0.6177413249, 0.6235457413, 0.6258548896, 0.6299053628, 0.6309326894, 0.6196196800, 0.6220706308, 0.6273830365, 0.6284213704, 0.6332158336, 0.6183299624, 0.6243744582, 0.6262698642, 0.6307425600, 0.6311234090, 0.6200000000, 0.6228998340, 0.6277255119, 0.6291975650, 0.6337751856, 0.6193319194, 0.6248250265, 0.6271155885, 0.6310074231, 0.6319110669, 0.6210992674, 0.6235685035, 0.6287203666, 0.6295942527, 0.6343293033, 0.6201447680, 0.6257172640, 0.6277143998, 0.6316209923, 0.6326650723, 0.6218564329, 0.6245562572, 0.6292085498, 0.6304603714, 0.6348529122, 0.6209508273, 0.6266963053, 0.6282738469, 0.6324686733, 0.6333301792, 0.6228268630, 0.6252526903, 0.6299925161, 0.6310075786, 0.6358336155, 0.6221994665, 0.6272534333, 0.6293840272, 0.6333066321, 0.6340148139, 0.6238084664, 0.6263303145, 0.6307871655, 0.6318815524, 0.6365111531, 0.6229126036, 0.6282453681, 0.6301499269, 0.6340885544, 0.6349320233, 0.6246604828, 0.6271496658, 0.6315643051, 0.6329992459, 0.6371596084, 0.6240190317, 0.6292133250, 0.6310010628, 0.6347736119, 0.6358462475, 0.6257021046, 0.6280638831, 0.6326791636, 0.6334574000, 0.6380596529, 0.6251318171, 0.6301115726, 0.6320839463, 0.6355469920, 0.6366038941, 0.6266275043, 0.6292953901, 0.6334099463, 0.6346750002, 0.6389118287, 0.6259938635, 0.6311477880, 0.6328054831, 0.6364704960, 0.6375586146, 0.6278436789, 0.6301418357, 0.6343899438, 0.6355564325, 0.6398616250, 0.6274045179, 0.6320878089, 0.6339208870, 0.6373802631, 0.6386554622, 0.6288578212, 0.6312210983, 0.6355008371, 0.6364213847, 0.6408627138, 0.6283774321, 0.6331443510, 0.6349442649, 0.6386478341, 0.6394124529, 0.6299783020, 0.6323780860, 0.6362577368, 0.6376626104, 0.6417436323, 0.6295866032, 0.6341900777, 0.6360073656, 0.6393189750, 0.6407561658, 0.6313668436, 0.6333934498, 0.6376372375, 0.6385854661, 0.6427383232, 0.6309191264, 0.6355015312, 0.6371919095, 0.6404291857, 0.6417107904, 0.6323280039, 0.6347894172, 0.6386394311, 0.6397035922, 0.6437882027, 0.6320768897, 0.6366612111, 0.6382227195, 0.6416379973, 0.6427362278, 0.6337119319, 0.6359649830, 0.6396408016, 0.6407894949, 0.6449936954, 0.6333924874, 0.6377706689, 0.6395336166, 0.6426430984, 0.6439997594, 0.6349452747, 0.6371102324, 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