The algebraic equation satisfied by the Probability Generating Function of L\ ife in a Casion where the stakes are from, -1, to , 1 By Shalosh B. Ekhad Theorem: Let a(n) be probability that you exit after EXACTLY n rounds in a \ casino where in each turn The probability that you win, 1, dollars is, p[1] The probability that you lose, -1, dollars is, p[-1] and the probability that you neither win nor lose is, p[0] Of course the probabilities have to add-up to 1, in other wrods p[-1] + p[0] + p[1] = 1 We also assume that the Casino stays in business, in other words, the expect\ ed gain in one round is negative: -p[-1] + p[1] < 1 Let X(t) be the ordinary generating function of that sequence, in other word\ s infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 X(t) satisifies the algebraic equation 2 p[1] t X(t) + (t p[0] - 1) X(t) + p[-1] t = 0 and in Maple notation p[1]*t*X(t)^2+(t*p[0]-1)*X(t)+p[-1]*t = 0 Or more usefully (for computing many terms) 2 X(t) = p[1] t X(t) + p[0] t X(t) + p[-1] t and in Maple notation X(t) = p[1]*t*X(t)^2+p[0]*t*X(t)+p[-1]*t The explicit expression for the expected duration in the casino is 1 - ----------------- p[0] + 2 p[1] - 1 and in Maple notation -1/(p[0]+2*p[1]-1) The explicit expression for the standard-deviation is / 2 2 \1/2 |p[0] + 4 p[0] p[1] + 4 p[1] - p[0] - 4 p[1]| |---------------------------------------------| | 3 | \ (p[0] + 2 p[1] - 1) / and in Maple notation ((p[0]^2+4*p[0]*p[1]+4*p[1]^2-p[0]-4*p[1])/(p[0]+2*p[1]-1)^3)^(1/2) The explicit expression for the, 3, -th scaled moment is 4 3 2 2 3 4 3 - (p[0] + 8 p[0] p[1] + 24 p[0] p[1] + 32 p[0] p[1] + 16 p[1] - 2 p[0] 2 2 3 2 - 16 p[0] p[1] - 40 p[0] p[1] - 32 p[1] + 2 p[0] + 4 p[0] p[1] / 2 / | 5 + 12 p[1] - 2 p[0] + 4 p[1] + 1) / |(p[0] + 2 p[1] - 1) / | \ / 2 2 \3/2\ |p[0] + 4 p[0] p[1] + 4 p[1] - p[0] - 4 p[1]| | |---------------------------------------------| | | 3 | | \ (p[0] + 2 p[1] - 1) / / and in Maple notation -(p[0]^4+8*p[0]^3*p[1]+24*p[0]^2*p[1]^2+32*p[0]*p[1]^3+16*p[1]^4-2*p[0]^3-16*p[ 0]^2*p[1]-40*p[0]*p[1]^2-32*p[1]^3+2*p[0]^2+4*p[0]*p[1]+12*p[1]^2-2*p[0]+4*p[1] +1)/(p[0]+2*p[1]-1)^5/((p[0]^2+4*p[0]*p[1]+4*p[1]^2-p[0]-4*p[1])/(p[0]+2*p[1]-1 )^3)^(3/2) The explicit expression for the, 4, -th scaled moment is 6 5 4 2 3 3 2 4 (p[0] + 12 p[0] p[1] + 60 p[0] p[1] + 160 p[0] p[1] + 240 p[0] p[1] 5 6 4 3 2 + 192 p[0] p[1] + 64 p[1] - 6 p[0] p[1] - 48 p[0] p[1] 2 3 4 5 4 3 - 144 p[0] p[1] - 192 p[0] p[1] - 96 p[1] - p[0] - 36 p[0] p[1] 2 2 3 4 3 2 - 156 p[0] p[1] - 224 p[0] p[1] - 96 p[1] - 7 p[0] - 6 p[0] p[1] 2 3 2 2 + 180 p[0] p[1] + 208 p[1] + 12 p[0] + 84 p[0] p[1] - 36 p[1] - 5 p[0] / 2 2 2 - 48 p[1]) / ((p[0] + 4 p[0] p[1] + 4 p[1] - p[0] - 4 p[1]) / (p[0] + 2 p[1] - 1)) and in Maple notation (p[0]^6+12*p[0]^5*p[1]+60*p[0]^4*p[1]^2+160*p[0]^3*p[1]^3+240*p[0]^2*p[1]^4+192 *p[0]*p[1]^5+64*p[1]^6-6*p[0]^4*p[1]-48*p[0]^3*p[1]^2-144*p[0]^2*p[1]^3-192*p[0 ]*p[1]^4-96*p[1]^5-p[0]^4-36*p[0]^3*p[1]-156*p[0]^2*p[1]^2-224*p[0]*p[1]^3-96*p [1]^4-7*p[0]^3-6*p[0]^2*p[1]+180*p[0]*p[1]^2+208*p[1]^3+12*p[0]^2+84*p[0]*p[1]-\ 36*p[1]^2-5*p[0]-48*p[1])/(p[0]^2+4*p[0]*p[1]+4*p[1]^2-p[0]-4*p[1])^2/(p[0]+2*p [1]-1) This took, 0.056, seconds. The algebraic equation satisfied by the Probability Generating Function of L\ ife in a Casion where the stakes are from, -1, to , 2 By Shalosh B. Ekhad Theorem: Let a(n) be probability that you exit after EXACTLY n rounds in a \ casino where in each turn The probability that you win, 1, dollars is, p[1] The probability that you win, 2, dollars is, p[2] The probability that you lose, -1, dollars is, p[-1] and the probability that you neither win nor lose is, p[0] Of course the probabilities have to add-up to 1, in other wrods p[-1] + p[0] + p[1] + p[2] = 1 We also assume that the Casino stays in business, in other words, the expect\ ed gain in one round is negative: -p[-1] + p[1] + 2 p[2] < 1 Let X(t) be the ordinary generating function of that sequence, in other word\ s infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 X(t) satisifies the algebraic equation 3 2 p[2] t X(t) + X(t) p[1] t + (t p[0] - 1) X(t) + p[-1] t = 0 and in Maple notation p[2]*t*X(t)^3+X(t)^2*p[1]*t+(t*p[0]-1)*X(t)+p[-1]*t = 0 Or more usefully (for computing many terms) 3 2 X(t) = p[2] t X(t) + X(t) p[1] t + p[0] t X(t) + p[-1] t and in Maple notation X(t) = p[2]*t*X(t)^3+X(t)^2*p[1]*t+p[0]*t*X(t)+p[-1]*t The explicit expression for the expected duration in the casino is 1 - -------------------------- p[0] + 2 p[1] + 3 p[2] - 1 and in Maple notation -1/(p[0]+2*p[1]+3*p[2]-1) The explicit expression for the standard-deviation is / 2 2 2 |(p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 4 p[1] + 12 p[1] p[2] + 9 p[2] - p[0] \ / 3\1/2 - 4 p[1] - 9 p[2]) / (p[0] + 2 p[1] + 3 p[2] - 1) | / / and in Maple notation ((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+4*p[1]^2+12*p[1]*p[2]+9*p[2]^2-p[0]-4*p[1]-9*p [2])/(p[0]+2*p[1]+3*p[2]-1)^3)^(1/2) The explicit expression for the, 3, -th scaled moment is 4 3 3 2 2 2 - (p[0] + 8 p[0] p[1] + 12 p[0] p[2] + 24 p[0] p[1] + 72 p[0] p[1] p[2] 2 2 3 2 + 54 p[0] p[2] + 32 p[0] p[1] + 144 p[0] p[1] p[2] 2 3 4 3 + 216 p[0] p[1] p[2] + 108 p[0] p[2] + 16 p[1] + 96 p[1] p[2] 2 2 3 4 3 2 + 216 p[1] p[2] + 216 p[1] p[2] + 81 p[2] - 2 p[0] - 16 p[0] p[1] 2 2 2 - 30 p[0] p[2] - 40 p[0] p[1] - 144 p[0] p[1] p[2] - 126 p[0] p[2] 3 2 2 3 2 - 32 p[1] - 168 p[1] p[2] - 288 p[1] p[2] - 162 p[2] + 2 p[0] 2 2 + 4 p[0] p[1] - 6 p[0] p[2] + 12 p[1] + 48 p[1] p[2] + 72 p[2] - 2 p[0] / / 5 / 2 + 4 p[1] + 24 p[2] + 1) / |(p[0] + 2 p[1] + 3 p[2] - 1) |(p[0] / \ \ 2 2 + 4 p[0] p[1] + 6 p[0] p[2] + 4 p[1] + 12 p[1] p[2] + 9 p[2] - p[0] / 3\3/2\ - 4 p[1] - 9 p[2]) / (p[0] + 2 p[1] + 3 p[2] - 1) | | / / / and in Maple notation -(p[0]^4+8*p[0]^3*p[1]+12*p[0]^3*p[2]+24*p[0]^2*p[1]^2+72*p[0]^2*p[1]*p[2]+54*p [0]^2*p[2]^2+32*p[0]*p[1]^3+144*p[0]*p[1]^2*p[2]+216*p[0]*p[1]*p[2]^2+108*p[0]* p[2]^3+16*p[1]^4+96*p[1]^3*p[2]+216*p[1]^2*p[2]^2+216*p[1]*p[2]^3+81*p[2]^4-2*p [0]^3-16*p[0]^2*p[1]-30*p[0]^2*p[2]-40*p[0]*p[1]^2-144*p[0]*p[1]*p[2]-126*p[0]* p[2]^2-32*p[1]^3-168*p[1]^2*p[2]-288*p[1]*p[2]^2-162*p[2]^3+2*p[0]^2+4*p[0]*p[1 ]-6*p[0]*p[2]+12*p[1]^2+48*p[1]*p[2]+72*p[2]^2-2*p[0]+4*p[1]+24*p[2]+1)/(p[0]+2 *p[1]+3*p[2]-1)^5/((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+4*p[1]^2+12*p[1]*p[2]+9*p[2] ^2-p[0]-4*p[1]-9*p[2])/(p[0]+2*p[1]+3*p[2]-1)^3)^(3/2) The explicit expression for the, 4, -th scaled moment is 6 5 5 4 2 4 (p[0] + 12 p[0] p[1] + 18 p[0] p[2] + 60 p[0] p[1] + 180 p[0] p[1] p[2] 4 2 3 3 3 2 + 135 p[0] p[2] + 160 p[0] p[1] + 720 p[0] p[1] p[2] 3 2 3 3 2 4 + 1080 p[0] p[1] p[2] + 540 p[0] p[2] + 240 p[0] p[1] 2 3 2 2 2 2 3 + 1440 p[0] p[1] p[2] + 3240 p[0] p[1] p[2] + 3240 p[0] p[1] p[2] 2 4 5 4 + 1215 p[0] p[2] + 192 p[0] p[1] + 1440 p[0] p[1] p[2] 3 2 2 3 4 + 4320 p[0] p[1] p[2] + 6480 p[0] p[1] p[2] + 4860 p[0] p[1] p[2] 5 6 5 4 2 + 1458 p[0] p[2] + 64 p[1] + 576 p[1] p[2] + 2160 p[1] p[2] 3 3 2 4 5 6 + 4320 p[1] p[2] + 4860 p[1] p[2] + 2916 p[1] p[2] + 729 p[2] 4 4 3 2 3 - 6 p[0] p[1] - 18 p[0] p[2] - 48 p[0] p[1] - 216 p[0] p[1] p[2] 3 2 2 3 2 2 - 216 p[0] p[2] - 144 p[0] p[1] - 864 p[0] p[1] p[2] 2 2 2 3 4 - 1620 p[0] p[1] p[2] - 972 p[0] p[2] - 192 p[0] p[1] 3 2 2 3 - 1440 p[0] p[1] p[2] - 3888 p[0] p[1] p[2] - 4536 p[0] p[1] p[2] 4 5 4 3 2 - 1944 p[0] p[2] - 96 p[1] - 864 p[1] p[2] - 3024 p[1] p[2] 2 3 4 5 4 3 - 5184 p[1] p[2] - 4374 p[1] p[2] - 1458 p[2] - p[0] - 36 p[0] p[1] 3 2 2 2 2 2 - 114 p[0] p[2] - 156 p[0] p[1] - 720 p[0] p[1] p[2] - 648 p[0] p[2] 3 2 2 - 224 p[0] p[1] - 1296 p[0] p[1] p[2] - 2052 p[0] p[1] p[2] 3 4 3 2 2 - 918 p[0] p[2] - 96 p[1] - 624 p[1] p[2] - 1188 p[1] p[2] 3 4 3 2 2 - 648 p[1] p[2] + 81 p[2] - 7 p[0] - 6 p[0] p[1] + 27 p[0] p[2] 2 2 3 + 180 p[0] p[1] + 1164 p[0] p[1] p[2] + 1791 p[0] p[2] + 208 p[1] 2 2 3 2 + 1500 p[1] p[2] + 3186 p[1] p[2] + 1701 p[2] + 12 p[0] + 84 p[0] p[1] 2 2 + 270 p[0] p[2] - 36 p[1] - 408 p[1] p[2] - 1062 p[2] - 5 p[0] - 48 p[1] / 2 2 - 183 p[2]) / ((p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 4 p[1] / 2 2 + 12 p[1] p[2] + 9 p[2] - p[0] - 4 p[1] - 9 p[2]) (p[0] + 2 p[1] + 3 p[2] - 1)) and in Maple notation (p[0]^6+12*p[0]^5*p[1]+18*p[0]^5*p[2]+60*p[0]^4*p[1]^2+180*p[0]^4*p[1]*p[2]+135 *p[0]^4*p[2]^2+160*p[0]^3*p[1]^3+720*p[0]^3*p[1]^2*p[2]+1080*p[0]^3*p[1]*p[2]^2 +540*p[0]^3*p[2]^3+240*p[0]^2*p[1]^4+1440*p[0]^2*p[1]^3*p[2]+3240*p[0]^2*p[1]^2 *p[2]^2+3240*p[0]^2*p[1]*p[2]^3+1215*p[0]^2*p[2]^4+192*p[0]*p[1]^5+1440*p[0]*p[ 1]^4*p[2]+4320*p[0]*p[1]^3*p[2]^2+6480*p[0]*p[1]^2*p[2]^3+4860*p[0]*p[1]*p[2]^4 +1458*p[0]*p[2]^5+64*p[1]^6+576*p[1]^5*p[2]+2160*p[1]^4*p[2]^2+4320*p[1]^3*p[2] ^3+4860*p[1]^2*p[2]^4+2916*p[1]*p[2]^5+729*p[2]^6-6*p[0]^4*p[1]-18*p[0]^4*p[2]-\ 48*p[0]^3*p[1]^2-216*p[0]^3*p[1]*p[2]-216*p[0]^3*p[2]^2-144*p[0]^2*p[1]^3-864*p [0]^2*p[1]^2*p[2]-1620*p[0]^2*p[1]*p[2]^2-972*p[0]^2*p[2]^3-192*p[0]*p[1]^4-\ 1440*p[0]*p[1]^3*p[2]-3888*p[0]*p[1]^2*p[2]^2-4536*p[0]*p[1]*p[2]^3-1944*p[0]*p [2]^4-96*p[1]^5-864*p[1]^4*p[2]-3024*p[1]^3*p[2]^2-5184*p[1]^2*p[2]^3-4374*p[1] *p[2]^4-1458*p[2]^5-p[0]^4-36*p[0]^3*p[1]-114*p[0]^3*p[2]-156*p[0]^2*p[1]^2-720 *p[0]^2*p[1]*p[2]-648*p[0]^2*p[2]^2-224*p[0]*p[1]^3-1296*p[0]*p[1]^2*p[2]-2052* p[0]*p[1]*p[2]^2-918*p[0]*p[2]^3-96*p[1]^4-624*p[1]^3*p[2]-1188*p[1]^2*p[2]^2-\ 648*p[1]*p[2]^3+81*p[2]^4-7*p[0]^3-6*p[0]^2*p[1]+27*p[0]^2*p[2]+180*p[0]*p[1]^2 +1164*p[0]*p[1]*p[2]+1791*p[0]*p[2]^2+208*p[1]^3+1500*p[1]^2*p[2]+3186*p[1]*p[2 ]^2+1701*p[2]^3+12*p[0]^2+84*p[0]*p[1]+270*p[0]*p[2]-36*p[1]^2-408*p[1]*p[2]-\ 1062*p[2]^2-5*p[0]-48*p[1]-183*p[2])/(p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+4*p[1]^2+ 12*p[1]*p[2]+9*p[2]^2-p[0]-4*p[1]-9*p[2])^2/(p[0]+2*p[1]+3*p[2]-1) This took, 0.071, seconds. The algebraic equation satisfied by the Probability Generating Function of L\ ife in a Casion where the stakes are from, -1, to , 3 By Shalosh B. Ekhad Theorem: Let a(n) be probability that you exit after EXACTLY n rounds in a \ casino where in each turn The probability that you win, 1, dollars is, p[1] The probability that you win, 2, dollars is, p[2] The probability that you win, 3, dollars is, p[3] The probability that you lose, -1, dollars is, p[-1] and the probability that you neither win nor lose is, p[0] Of course the probabilities have to add-up to 1, in other wrods p[-1] + p[0] + p[1] + p[2] + p[3] = 1 We also assume that the Casino stays in business, in other words, the expect\ ed gain in one round is negative: -p[-1] + p[1] + 2 p[2] + 3 p[3] < 1 Let X(t) be the ordinary generating function of that sequence, in other word\ s infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 X(t) satisifies the algebraic equation 4 3 2 p[3] t X(t) + X(t) p[2] t + X(t) p[1] t + (t p[0] - 1) X(t) + p[-1] t = 0 and in Maple notation p[3]*t*X(t)^4+X(t)^3*p[2]*t+X(t)^2*p[1]*t+(t*p[0]-1)*X(t)+p[-1]*t = 0 Or more usefully (for computing many terms) 4 3 2 X(t) = p[3] t X(t) + X(t) p[2] t + X(t) p[1] t + p[0] t X(t) + p[-1] t and in Maple notation X(t) = p[3]*t*X(t)^4+X(t)^3*p[2]*t+X(t)^2*p[1]*t+p[0]*t*X(t)+p[-1]*t The explicit expression for the expected duration in the casino is 1 - ----------------------------------- p[0] + 2 p[1] + 3 p[2] + 4 p[3] - 1 and in Maple notation -1/(p[0]+2*p[1]+3*p[2]+4*p[3]-1) The explicit expression for the standard-deviation is / 2 2 |(p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] + 4 p[1] + 12 p[1] p[2] \ 2 2 + 16 p[1] p[3] + 9 p[2] + 24 p[2] p[3] + 16 p[3] - p[0] - 4 p[1] / 3\1/2 - 9 p[2] - 16 p[3]) / (p[0] + 2 p[1] + 3 p[2] + 4 p[3] - 1) | / / and in Maple notation ((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+4*p[1]^2+12*p[1]*p[2]+16*p[1]*p[3] +9*p[2]^2+24*p[2]*p[3]+16*p[3]^2-p[0]-4*p[1]-9*p[2]-16*p[3])/(p[0]+2*p[1]+3*p[2 ]+4*p[3]-1)^3)^(1/2) The explicit expression for the, 3, -th scaled moment is 4 3 3 3 2 2 - (p[0] + 8 p[0] p[1] + 12 p[0] p[2] + 16 p[0] p[3] + 24 p[0] p[1] 2 2 2 2 + 72 p[0] p[1] p[2] + 96 p[0] p[1] p[3] + 54 p[0] p[2] 2 2 2 3 + 144 p[0] p[2] p[3] + 96 p[0] p[3] + 32 p[0] p[1] 2 2 2 + 144 p[0] p[1] p[2] + 192 p[0] p[1] p[3] + 216 p[0] p[1] p[2] 2 3 + 576 p[0] p[1] p[2] p[3] + 384 p[0] p[1] p[3] + 108 p[0] p[2] 2 2 3 4 + 432 p[0] p[2] p[3] + 576 p[0] p[2] p[3] + 256 p[0] p[3] + 16 p[1] 3 3 2 2 2 + 96 p[1] p[2] + 128 p[1] p[3] + 216 p[1] p[2] + 576 p[1] p[2] p[3] 2 2 3 2 + 384 p[1] p[3] + 216 p[1] p[2] + 864 p[1] p[2] p[3] 2 3 4 3 + 1152 p[1] p[2] p[3] + 512 p[1] p[3] + 81 p[2] + 432 p[2] p[3] 2 2 3 4 3 2 + 864 p[2] p[3] + 768 p[2] p[3] + 256 p[3] - 2 p[0] - 16 p[0] p[1] 2 2 2 - 30 p[0] p[2] - 48 p[0] p[3] - 40 p[0] p[1] - 144 p[0] p[1] p[2] 2 - 224 p[0] p[1] p[3] - 126 p[0] p[2] - 384 p[0] p[2] p[3] 2 3 2 2 - 288 p[0] p[3] - 32 p[1] - 168 p[1] p[2] - 256 p[1] p[3] 2 2 3 - 288 p[1] p[2] - 864 p[1] p[2] p[3] - 640 p[1] p[3] - 162 p[2] 2 2 3 2 - 720 p[2] p[3] - 1056 p[2] p[3] - 512 p[3] + 2 p[0] + 4 p[0] p[1] 2 - 6 p[0] p[2] - 32 p[0] p[3] + 12 p[1] + 48 p[1] p[2] + 64 p[1] p[3] 2 2 + 72 p[2] + 264 p[2] p[3] + 272 p[3] - 2 p[0] + 4 p[1] + 24 p[2] / / 5 / 2 + 64 p[3] + 1) / |(p[0] + 2 p[1] + 3 p[2] + 4 p[3] - 1) |(p[0] / \ \ 2 + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] + 4 p[1] + 12 p[1] p[2] 2 2 + 16 p[1] p[3] + 9 p[2] + 24 p[2] p[3] + 16 p[3] - p[0] - 4 p[1] / 3\3/2\ - 9 p[2] - 16 p[3]) / (p[0] + 2 p[1] + 3 p[2] + 4 p[3] - 1) | | / / / and in Maple notation -(p[0]^4+8*p[0]^3*p[1]+12*p[0]^3*p[2]+16*p[0]^3*p[3]+24*p[0]^2*p[1]^2+72*p[0]^2 *p[1]*p[2]+96*p[0]^2*p[1]*p[3]+54*p[0]^2*p[2]^2+144*p[0]^2*p[2]*p[3]+96*p[0]^2* p[3]^2+32*p[0]*p[1]^3+144*p[0]*p[1]^2*p[2]+192*p[0]*p[1]^2*p[3]+216*p[0]*p[1]*p [2]^2+576*p[0]*p[1]*p[2]*p[3]+384*p[0]*p[1]*p[3]^2+108*p[0]*p[2]^3+432*p[0]*p[2 ]^2*p[3]+576*p[0]*p[2]*p[3]^2+256*p[0]*p[3]^3+16*p[1]^4+96*p[1]^3*p[2]+128*p[1] ^3*p[3]+216*p[1]^2*p[2]^2+576*p[1]^2*p[2]*p[3]+384*p[1]^2*p[3]^2+216*p[1]*p[2]^ 3+864*p[1]*p[2]^2*p[3]+1152*p[1]*p[2]*p[3]^2+512*p[1]*p[3]^3+81*p[2]^4+432*p[2] ^3*p[3]+864*p[2]^2*p[3]^2+768*p[2]*p[3]^3+256*p[3]^4-2*p[0]^3-16*p[0]^2*p[1]-30 *p[0]^2*p[2]-48*p[0]^2*p[3]-40*p[0]*p[1]^2-144*p[0]*p[1]*p[2]-224*p[0]*p[1]*p[3 ]-126*p[0]*p[2]^2-384*p[0]*p[2]*p[3]-288*p[0]*p[3]^2-32*p[1]^3-168*p[1]^2*p[2]-\ 256*p[1]^2*p[3]-288*p[1]*p[2]^2-864*p[1]*p[2]*p[3]-640*p[1]*p[3]^2-162*p[2]^3-\ 720*p[2]^2*p[3]-1056*p[2]*p[3]^2-512*p[3]^3+2*p[0]^2+4*p[0]*p[1]-6*p[0]*p[2]-32 *p[0]*p[3]+12*p[1]^2+48*p[1]*p[2]+64*p[1]*p[3]+72*p[2]^2+264*p[2]*p[3]+272*p[3] ^2-2*p[0]+4*p[1]+24*p[2]+64*p[3]+1)/(p[0]+2*p[1]+3*p[2]+4*p[3]-1)^5/((p[0]^2+4* p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+4*p[1]^2+12*p[1]*p[2]+16*p[1]*p[3]+9*p[2]^2+ 24*p[2]*p[3]+16*p[3]^2-p[0]-4*p[1]-9*p[2]-16*p[3])/(p[0]+2*p[1]+3*p[2]+4*p[3]-1 )^3)^(3/2) The explicit expression for the, 4, -th scaled moment is 6 5 5 5 4 2 (p[0] + 12 p[0] p[1] + 18 p[0] p[2] + 24 p[0] p[3] + 60 p[0] p[1] 4 4 4 2 + 180 p[0] p[1] p[2] + 240 p[0] p[1] p[3] + 135 p[0] p[2] 4 4 2 3 3 + 360 p[0] p[2] p[3] + 240 p[0] p[3] + 160 p[0] p[1] 3 2 3 2 3 2 + 720 p[0] p[1] p[2] + 960 p[0] p[1] p[3] + 1080 p[0] p[1] p[2] 3 3 2 3 3 + 2880 p[0] p[1] p[2] p[3] + 1920 p[0] p[1] p[3] + 540 p[0] p[2] 3 2 3 2 3 3 + 2160 p[0] p[2] p[3] + 2880 p[0] p[2] p[3] + 1280 p[0] p[3] 2 4 2 3 2 3 + 240 p[0] p[1] + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] 2 2 2 2 2 + 3240 p[0] p[1] p[2] + 8640 p[0] p[1] p[2] p[3] 2 2 2 2 3 + 5760 p[0] p[1] p[3] + 3240 p[0] p[1] p[2] 2 2 2 2 + 12960 p[0] p[1] p[2] p[3] + 17280 p[0] p[1] p[2] p[3] 2 3 2 4 2 3 + 7680 p[0] p[1] p[3] + 1215 p[0] p[2] + 6480 p[0] p[2] p[3] 2 2 2 2 3 2 4 + 12960 p[0] p[2] p[3] + 11520 p[0] p[2] p[3] + 3840 p[0] p[3] 5 4 4 + 192 p[0] p[1] + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] 3 2 3 + 4320 p[0] p[1] p[2] + 11520 p[0] p[1] p[2] p[3] 3 2 2 3 + 7680 p[0] p[1] p[3] + 6480 p[0] p[1] p[2] 2 2 2 2 + 25920 p[0] p[1] p[2] p[3] + 34560 p[0] p[1] p[2] p[3] 2 3 4 + 15360 p[0] p[1] p[3] + 4860 p[0] p[1] p[2] 3 2 2 + 25920 p[0] p[1] p[2] p[3] + 51840 p[0] p[1] p[2] p[3] 3 4 5 + 46080 p[0] p[1] p[2] p[3] + 15360 p[0] p[1] p[3] + 1458 p[0] p[2] 4 3 2 2 3 + 9720 p[0] p[2] p[3] + 25920 p[0] p[2] p[3] + 34560 p[0] p[2] p[3] 4 5 6 5 + 23040 p[0] p[2] p[3] + 6144 p[0] p[3] + 64 p[1] + 576 p[1] p[2] 5 4 2 4 + 768 p[1] p[3] + 2160 p[1] p[2] + 5760 p[1] p[2] p[3] 4 2 3 3 3 2 + 3840 p[1] p[3] + 4320 p[1] p[2] + 17280 p[1] p[2] p[3] 3 2 3 3 2 4 + 23040 p[1] p[2] p[3] + 10240 p[1] p[3] + 4860 p[1] p[2] 2 3 2 2 2 + 25920 p[1] p[2] p[3] + 51840 p[1] p[2] p[3] 2 3 2 4 5 + 46080 p[1] p[2] p[3] + 15360 p[1] p[3] + 2916 p[1] p[2] 4 3 2 2 3 + 19440 p[1] p[2] p[3] + 51840 p[1] p[2] p[3] + 69120 p[1] p[2] p[3] 4 5 6 5 + 46080 p[1] p[2] p[3] + 12288 p[1] p[3] + 729 p[2] + 5832 p[2] p[3] 4 2 3 3 2 4 + 19440 p[2] p[3] + 34560 p[2] p[3] + 34560 p[2] p[3] 5 6 4 4 + 18432 p[2] p[3] + 4096 p[3] - 6 p[0] p[1] - 18 p[0] p[2] 4 3 2 3 - 36 p[0] p[3] - 48 p[0] p[1] - 216 p[0] p[1] p[2] 3 3 2 3 - 384 p[0] p[1] p[3] - 216 p[0] p[2] - 720 p[0] p[2] p[3] 3 2 2 3 2 2 - 576 p[0] p[3] - 144 p[0] p[1] - 864 p[0] p[1] p[2] 2 2 2 2 - 1440 p[0] p[1] p[3] - 1620 p[0] p[1] p[2] 2 2 2 2 3 - 5184 p[0] p[1] p[2] p[3] - 4032 p[0] p[1] p[3] - 972 p[0] p[2] 2 2 2 2 2 3 - 4536 p[0] p[2] p[3] - 6912 p[0] p[2] p[3] - 3456 p[0] p[3] 4 3 3 - 192 p[0] p[1] - 1440 p[0] p[1] p[2] - 2304 p[0] p[1] p[3] 2 2 2 - 3888 p[0] p[1] p[2] - 12096 p[0] p[1] p[2] p[3] 2 2 3 - 9216 p[0] p[1] p[3] - 4536 p[0] p[1] p[2] 2 2 - 20736 p[0] p[1] p[2] p[3] - 31104 p[0] p[1] p[2] p[3] 3 4 3 - 15360 p[0] p[1] p[3] - 1944 p[0] p[2] - 11664 p[0] p[2] p[3] 2 2 3 4 - 25920 p[0] p[2] p[3] - 25344 p[0] p[2] p[3] - 9216 p[0] p[3] 5 4 4 3 2 - 96 p[1] - 864 p[1] p[2] - 1344 p[1] p[3] - 3024 p[1] p[2] 3 3 2 2 3 - 9216 p[1] p[2] p[3] - 6912 p[1] p[3] - 5184 p[1] p[2] 2 2 2 2 2 3 - 23328 p[1] p[2] p[3] - 34560 p[1] p[2] p[3] - 16896 p[1] p[3] 4 3 2 2 - 4374 p[1] p[2] - 25920 p[1] p[2] p[3] - 57024 p[1] p[2] p[3] 3 4 5 4 - 55296 p[1] p[2] p[3] - 19968 p[1] p[3] - 1458 p[2] - 10692 p[2] p[3] 3 2 2 3 4 5 - 31104 p[2] p[3] - 44928 p[2] p[3] - 32256 p[2] p[3] - 9216 p[3] 4 3 3 3 - p[0] - 36 p[0] p[1] - 114 p[0] p[2] - 256 p[0] p[3] 2 2 2 2 - 156 p[0] p[1] - 720 p[0] p[1] p[2] - 1440 p[0] p[1] p[3] 2 2 2 2 2 - 648 p[0] p[2] - 2232 p[0] p[2] p[3] - 1680 p[0] p[3] 3 2 2 - 224 p[0] p[1] - 1296 p[0] p[1] p[2] - 2400 p[0] p[1] p[3] 2 2 - 2052 p[0] p[1] p[2] - 6624 p[0] p[1] p[2] p[3] - 4608 p[0] p[1] p[3] 3 2 2 - 918 p[0] p[2] - 3888 p[0] p[2] p[3] - 4608 p[0] p[2] p[3] 3 4 3 3 - 1408 p[0] p[3] - 96 p[1] - 624 p[1] p[2] - 1088 p[1] p[3] 2 2 2 2 2 - 1188 p[1] p[2] - 3456 p[1] p[2] p[3] - 1920 p[1] p[3] 3 2 2 - 648 p[1] p[2] - 1728 p[1] p[2] p[3] + 576 p[1] p[2] p[3] 3 4 3 2 2 + 2304 p[1] p[3] + 81 p[2] + 1512 p[2] p[3] + 6480 p[2] p[3] 3 4 3 2 2 + 9984 p[2] p[3] + 5120 p[3] - 7 p[0] - 6 p[0] p[1] + 27 p[0] p[2] 2 2 + 36 p[0] p[3] + 180 p[0] p[1] + 1164 p[0] p[1] p[2] 2 + 2352 p[0] p[1] p[3] + 1791 p[0] p[2] + 7416 p[0] p[2] p[3] 2 3 2 2 + 7824 p[0] p[3] + 208 p[1] + 1500 p[1] p[2] + 2880 p[1] p[3] 2 2 3 + 3186 p[1] p[2] + 11856 p[1] p[2] p[3] + 10944 p[1] p[3] + 1701 p[2] 2 2 3 2 + 8244 p[2] p[3] + 11952 p[2] p[3] + 4352 p[3] + 12 p[0] 2 + 84 p[0] p[1] + 270 p[0] p[2] + 720 p[0] p[3] - 36 p[1] - 408 p[1] p[2] 2 2 - 768 p[1] p[3] - 1062 p[2] - 4824 p[2] p[3] - 5808 p[3] - 5 p[0] / 2 - 48 p[1] - 183 p[2] - 488 p[3]) / ((p[0] + 4 p[0] p[1] + 6 p[0] p[2] / 2 2 + 8 p[0] p[3] + 4 p[1] + 12 p[1] p[2] + 16 p[1] p[3] + 9 p[2] 2 2 + 24 p[2] p[3] + 16 p[3] - p[0] - 4 p[1] - 9 p[2] - 16 p[3]) (p[0] + 2 p[1] + 3 p[2] + 4 p[3] - 1)) and in Maple notation (p[0]^6+12*p[0]^5*p[1]+18*p[0]^5*p[2]+24*p[0]^5*p[3]+60*p[0]^4*p[1]^2+180*p[0]^ 4*p[1]*p[2]+240*p[0]^4*p[1]*p[3]+135*p[0]^4*p[2]^2+360*p[0]^4*p[2]*p[3]+240*p[0 ]^4*p[3]^2+160*p[0]^3*p[1]^3+720*p[0]^3*p[1]^2*p[2]+960*p[0]^3*p[1]^2*p[3]+1080 *p[0]^3*p[1]*p[2]^2+2880*p[0]^3*p[1]*p[2]*p[3]+1920*p[0]^3*p[1]*p[3]^2+540*p[0] ^3*p[2]^3+2160*p[0]^3*p[2]^2*p[3]+2880*p[0]^3*p[2]*p[3]^2+1280*p[0]^3*p[3]^3+ 240*p[0]^2*p[1]^4+1440*p[0]^2*p[1]^3*p[2]+1920*p[0]^2*p[1]^3*p[3]+3240*p[0]^2*p [1]^2*p[2]^2+8640*p[0]^2*p[1]^2*p[2]*p[3]+5760*p[0]^2*p[1]^2*p[3]^2+3240*p[0]^2 *p[1]*p[2]^3+12960*p[0]^2*p[1]*p[2]^2*p[3]+17280*p[0]^2*p[1]*p[2]*p[3]^2+7680*p [0]^2*p[1]*p[3]^3+1215*p[0]^2*p[2]^4+6480*p[0]^2*p[2]^3*p[3]+12960*p[0]^2*p[2]^ 2*p[3]^2+11520*p[0]^2*p[2]*p[3]^3+3840*p[0]^2*p[3]^4+192*p[0]*p[1]^5+1440*p[0]* p[1]^4*p[2]+1920*p[0]*p[1]^4*p[3]+4320*p[0]*p[1]^3*p[2]^2+11520*p[0]*p[1]^3*p[2 ]*p[3]+7680*p[0]*p[1]^3*p[3]^2+6480*p[0]*p[1]^2*p[2]^3+25920*p[0]*p[1]^2*p[2]^2 *p[3]+34560*p[0]*p[1]^2*p[2]*p[3]^2+15360*p[0]*p[1]^2*p[3]^3+4860*p[0]*p[1]*p[2 ]^4+25920*p[0]*p[1]*p[2]^3*p[3]+51840*p[0]*p[1]*p[2]^2*p[3]^2+46080*p[0]*p[1]*p [2]*p[3]^3+15360*p[0]*p[1]*p[3]^4+1458*p[0]*p[2]^5+9720*p[0]*p[2]^4*p[3]+25920* p[0]*p[2]^3*p[3]^2+34560*p[0]*p[2]^2*p[3]^3+23040*p[0]*p[2]*p[3]^4+6144*p[0]*p[ 3]^5+64*p[1]^6+576*p[1]^5*p[2]+768*p[1]^5*p[3]+2160*p[1]^4*p[2]^2+5760*p[1]^4*p [2]*p[3]+3840*p[1]^4*p[3]^2+4320*p[1]^3*p[2]^3+17280*p[1]^3*p[2]^2*p[3]+23040*p [1]^3*p[2]*p[3]^2+10240*p[1]^3*p[3]^3+4860*p[1]^2*p[2]^4+25920*p[1]^2*p[2]^3*p[ 3]+51840*p[1]^2*p[2]^2*p[3]^2+46080*p[1]^2*p[2]*p[3]^3+15360*p[1]^2*p[3]^4+2916 *p[1]*p[2]^5+19440*p[1]*p[2]^4*p[3]+51840*p[1]*p[2]^3*p[3]^2+69120*p[1]*p[2]^2* p[3]^3+46080*p[1]*p[2]*p[3]^4+12288*p[1]*p[3]^5+729*p[2]^6+5832*p[2]^5*p[3]+ 19440*p[2]^4*p[3]^2+34560*p[2]^3*p[3]^3+34560*p[2]^2*p[3]^4+18432*p[2]*p[3]^5+ 4096*p[3]^6-6*p[0]^4*p[1]-18*p[0]^4*p[2]-36*p[0]^4*p[3]-48*p[0]^3*p[1]^2-216*p[ 0]^3*p[1]*p[2]-384*p[0]^3*p[1]*p[3]-216*p[0]^3*p[2]^2-720*p[0]^3*p[2]*p[3]-576* p[0]^3*p[3]^2-144*p[0]^2*p[1]^3-864*p[0]^2*p[1]^2*p[2]-1440*p[0]^2*p[1]^2*p[3]-\ 1620*p[0]^2*p[1]*p[2]^2-5184*p[0]^2*p[1]*p[2]*p[3]-4032*p[0]^2*p[1]*p[3]^2-972* p[0]^2*p[2]^3-4536*p[0]^2*p[2]^2*p[3]-6912*p[0]^2*p[2]*p[3]^2-3456*p[0]^2*p[3]^ 3-192*p[0]*p[1]^4-1440*p[0]*p[1]^3*p[2]-2304*p[0]*p[1]^3*p[3]-3888*p[0]*p[1]^2* p[2]^2-12096*p[0]*p[1]^2*p[2]*p[3]-9216*p[0]*p[1]^2*p[3]^2-4536*p[0]*p[1]*p[2]^ 3-20736*p[0]*p[1]*p[2]^2*p[3]-31104*p[0]*p[1]*p[2]*p[3]^2-15360*p[0]*p[1]*p[3]^ 3-1944*p[0]*p[2]^4-11664*p[0]*p[2]^3*p[3]-25920*p[0]*p[2]^2*p[3]^2-25344*p[0]*p [2]*p[3]^3-9216*p[0]*p[3]^4-96*p[1]^5-864*p[1]^4*p[2]-1344*p[1]^4*p[3]-3024*p[1 ]^3*p[2]^2-9216*p[1]^3*p[2]*p[3]-6912*p[1]^3*p[3]^2-5184*p[1]^2*p[2]^3-23328*p[ 1]^2*p[2]^2*p[3]-34560*p[1]^2*p[2]*p[3]^2-16896*p[1]^2*p[3]^3-4374*p[1]*p[2]^4-\ 25920*p[1]*p[2]^3*p[3]-57024*p[1]*p[2]^2*p[3]^2-55296*p[1]*p[2]*p[3]^3-19968*p[ 1]*p[3]^4-1458*p[2]^5-10692*p[2]^4*p[3]-31104*p[2]^3*p[3]^2-44928*p[2]^2*p[3]^3 -32256*p[2]*p[3]^4-9216*p[3]^5-p[0]^4-36*p[0]^3*p[1]-114*p[0]^3*p[2]-256*p[0]^3 *p[3]-156*p[0]^2*p[1]^2-720*p[0]^2*p[1]*p[2]-1440*p[0]^2*p[1]*p[3]-648*p[0]^2*p [2]^2-2232*p[0]^2*p[2]*p[3]-1680*p[0]^2*p[3]^2-224*p[0]*p[1]^3-1296*p[0]*p[1]^2 *p[2]-2400*p[0]*p[1]^2*p[3]-2052*p[0]*p[1]*p[2]^2-6624*p[0]*p[1]*p[2]*p[3]-4608 *p[0]*p[1]*p[3]^2-918*p[0]*p[2]^3-3888*p[0]*p[2]^2*p[3]-4608*p[0]*p[2]*p[3]^2-\ 1408*p[0]*p[3]^3-96*p[1]^4-624*p[1]^3*p[2]-1088*p[1]^3*p[3]-1188*p[1]^2*p[2]^2-\ 3456*p[1]^2*p[2]*p[3]-1920*p[1]^2*p[3]^2-648*p[1]*p[2]^3-1728*p[1]*p[2]^2*p[3]+ 576*p[1]*p[2]*p[3]^2+2304*p[1]*p[3]^3+81*p[2]^4+1512*p[2]^3*p[3]+6480*p[2]^2*p[ 3]^2+9984*p[2]*p[3]^3+5120*p[3]^4-7*p[0]^3-6*p[0]^2*p[1]+27*p[0]^2*p[2]+36*p[0] ^2*p[3]+180*p[0]*p[1]^2+1164*p[0]*p[1]*p[2]+2352*p[0]*p[1]*p[3]+1791*p[0]*p[2]^ 2+7416*p[0]*p[2]*p[3]+7824*p[0]*p[3]^2+208*p[1]^3+1500*p[1]^2*p[2]+2880*p[1]^2* p[3]+3186*p[1]*p[2]^2+11856*p[1]*p[2]*p[3]+10944*p[1]*p[3]^2+1701*p[2]^3+8244*p [2]^2*p[3]+11952*p[2]*p[3]^2+4352*p[3]^3+12*p[0]^2+84*p[0]*p[1]+270*p[0]*p[2]+ 720*p[0]*p[3]-36*p[1]^2-408*p[1]*p[2]-768*p[1]*p[3]-1062*p[2]^2-4824*p[2]*p[3]-\ 5808*p[3]^2-5*p[0]-48*p[1]-183*p[2]-488*p[3])/(p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8 *p[0]*p[3]+4*p[1]^2+12*p[1]*p[2]+16*p[1]*p[3]+9*p[2]^2+24*p[2]*p[3]+16*p[3]^2-p [0]-4*p[1]-9*p[2]-16*p[3])^2/(p[0]+2*p[1]+3*p[2]+4*p[3]-1) This took, 0.154, seconds. The algebraic equation satisfied by the Probability Generating Function of L\ ife in a Casion where the stakes are from, -1, to , 4 By Shalosh B. Ekhad Theorem: Let a(n) be probability that you exit after EXACTLY n rounds in a \ casino where in each turn The probability that you win, 1, dollars is, p[1] The probability that you win, 2, dollars is, p[2] The probability that you win, 3, dollars is, p[3] The probability that you win, 4, dollars is, p[4] The probability that you lose, -1, dollars is, p[-1] and the probability that you neither win nor lose is, p[0] Of course the probabilities have to add-up to 1, in other wrods p[-1] + p[0] + p[1] + p[2] + p[3] + p[4] = 1 We also assume that the Casino stays in business, in other words, the expect\ ed gain in one round is negative: -p[-1] + p[1] + 2 p[2] + 3 p[3] + 4 p[4] < 1 Let X(t) be the ordinary generating function of that sequence, in other word\ s infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 X(t) satisifies the algebraic equation 5 4 3 2 p[4] t X(t) + X(t) p[3] t + X(t) p[2] t + X(t) p[1] t + (t p[0] - 1) X(t) + p[-1] t = 0 and in Maple notation p[4]*t*X(t)^5+X(t)^4*p[3]*t+X(t)^3*p[2]*t+X(t)^2*p[1]*t+(t*p[0]-1)*X(t)+p[-1]*t = 0 Or more usefully (for computing many terms) 5 4 3 2 X(t) = p[4] t X(t) + X(t) p[3] t + X(t) p[2] t + X(t) p[1] t + p[0] t X(t) + p[-1] t and in Maple notation X(t) = p[4]*t*X(t)^5+X(t)^4*p[3]*t+X(t)^3*p[2]*t+X(t)^2*p[1]*t+p[0]*t*X(t)+p[-1 ]*t The explicit expression for the expected duration in the casino is 1 - -------------------------------------------- p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] - 1 and in Maple notation -1/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]-1) The explicit expression for the standard-deviation is / 2 2 |(p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] + 10 p[0] p[4] + 4 p[1] \ 2 + 12 p[1] p[2] + 16 p[1] p[3] + 20 p[1] p[4] + 9 p[2] + 24 p[2] p[3] 2 2 + 30 p[2] p[4] + 16 p[3] + 40 p[3] p[4] + 25 p[4] - p[0] - 4 p[1] / - 9 p[2] - 16 p[3] - 25 p[4]) / / 3\1/2 (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] - 1) | / and in Maple notation ((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+10*p[0]*p[4]+4*p[1]^2+12*p[1]*p[2] +16*p[1]*p[3]+20*p[1]*p[4]+9*p[2]^2+24*p[2]*p[3]+30*p[2]*p[4]+16*p[3]^2+40*p[3] *p[4]+25*p[4]^2-p[0]-4*p[1]-9*p[2]-16*p[3]-25*p[4])/(p[0]+2*p[1]+3*p[2]+4*p[3]+ 5*p[4]-1)^3)^(1/2) The explicit expression for the, 3, -th scaled moment is 4 3 3 3 3 - (p[0] + 8 p[0] p[1] + 12 p[0] p[2] + 16 p[0] p[3] + 20 p[0] p[4] 2 2 2 2 + 24 p[0] p[1] + 72 p[0] p[1] p[2] + 96 p[0] p[1] p[3] 2 2 2 2 + 120 p[0] p[1] p[4] + 54 p[0] p[2] + 144 p[0] p[2] p[3] 2 2 2 2 + 180 p[0] p[2] p[4] + 96 p[0] p[3] + 240 p[0] p[3] p[4] 2 2 3 2 + 150 p[0] p[4] + 32 p[0] p[1] + 144 p[0] p[1] p[2] 2 2 2 + 192 p[0] p[1] p[3] + 240 p[0] p[1] p[4] + 216 p[0] p[1] p[2] 2 + 576 p[0] p[1] p[2] p[3] + 720 p[0] p[1] p[2] p[4] + 384 p[0] p[1] p[3] 2 3 + 960 p[0] p[1] p[3] p[4] + 600 p[0] p[1] p[4] + 108 p[0] p[2] 2 2 2 + 432 p[0] p[2] p[3] + 540 p[0] p[2] p[4] + 576 p[0] p[2] p[3] 2 3 + 1440 p[0] p[2] p[3] p[4] + 900 p[0] p[2] p[4] + 256 p[0] p[3] 2 2 3 4 + 960 p[0] p[3] p[4] + 1200 p[0] p[3] p[4] + 500 p[0] p[4] + 16 p[1] 3 3 3 2 2 + 96 p[1] p[2] + 128 p[1] p[3] + 160 p[1] p[4] + 216 p[1] p[2] 2 2 2 2 + 576 p[1] p[2] p[3] + 720 p[1] p[2] p[4] + 384 p[1] p[3] 2 2 2 3 + 960 p[1] p[3] p[4] + 600 p[1] p[4] + 216 p[1] p[2] 2 2 2 + 864 p[1] p[2] p[3] + 1080 p[1] p[2] p[4] + 1152 p[1] p[2] p[3] 2 3 + 2880 p[1] p[2] p[3] p[4] + 1800 p[1] p[2] p[4] + 512 p[1] p[3] 2 2 3 4 + 1920 p[1] p[3] p[4] + 2400 p[1] p[3] p[4] + 1000 p[1] p[4] + 81 p[2] 3 3 2 2 2 + 432 p[2] p[3] + 540 p[2] p[4] + 864 p[2] p[3] + 2160 p[2] p[3] p[4] 2 2 3 2 + 1350 p[2] p[4] + 768 p[2] p[3] + 2880 p[2] p[3] p[4] 2 3 4 3 + 3600 p[2] p[3] p[4] + 1500 p[2] p[4] + 256 p[3] + 1280 p[3] p[4] 2 2 3 4 3 2 + 2400 p[3] p[4] + 2000 p[3] p[4] + 625 p[4] - 2 p[0] - 16 p[0] p[1] 2 2 2 2 - 30 p[0] p[2] - 48 p[0] p[3] - 70 p[0] p[4] - 40 p[0] p[1] - 144 p[0] p[1] p[2] - 224 p[0] p[1] p[3] - 320 p[0] p[1] p[4] 2 - 126 p[0] p[2] - 384 p[0] p[2] p[3] - 540 p[0] p[2] p[4] 2 2 3 - 288 p[0] p[3] - 800 p[0] p[3] p[4] - 550 p[0] p[4] - 32 p[1] 2 2 2 2 - 168 p[1] p[2] - 256 p[1] p[3] - 360 p[1] p[4] - 288 p[1] p[2] 2 - 864 p[1] p[2] p[3] - 1200 p[1] p[2] p[4] - 640 p[1] p[3] 2 3 2 - 1760 p[1] p[3] p[4] - 1200 p[1] p[4] - 162 p[2] - 720 p[2] p[3] 2 2 2 - 990 p[2] p[4] - 1056 p[2] p[3] - 2880 p[2] p[3] p[4] - 1950 p[2] p[4] 3 2 2 3 2 - 512 p[3] - 2080 p[3] p[4] - 2800 p[3] p[4] - 1250 p[4] + 2 p[0] 2 + 4 p[0] p[1] - 6 p[0] p[2] - 32 p[0] p[3] - 80 p[0] p[4] + 12 p[1] 2 + 48 p[1] p[2] + 64 p[1] p[3] + 60 p[1] p[4] + 72 p[2] + 264 p[2] p[3] 2 2 + 390 p[2] p[4] + 272 p[3] + 880 p[3] p[4] + 750 p[4] - 2 p[0] + 4 p[1] / / + 24 p[2] + 64 p[3] + 130 p[4] + 1) / | / \ 5 / 2 (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] - 1) |(p[0] + 4 p[0] p[1] \ 2 + 6 p[0] p[2] + 8 p[0] p[3] + 10 p[0] p[4] + 4 p[1] + 12 p[1] p[2] 2 + 16 p[1] p[3] + 20 p[1] p[4] + 9 p[2] + 24 p[2] p[3] + 30 p[2] p[4] 2 2 + 16 p[3] + 40 p[3] p[4] + 25 p[4] - p[0] - 4 p[1] - 9 p[2] - 16 p[3] / 3\3/2\ - 25 p[4]) / (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] - 1) | | / / / and in Maple notation -(p[0]^4+8*p[0]^3*p[1]+12*p[0]^3*p[2]+16*p[0]^3*p[3]+20*p[0]^3*p[4]+24*p[0]^2*p [1]^2+72*p[0]^2*p[1]*p[2]+96*p[0]^2*p[1]*p[3]+120*p[0]^2*p[1]*p[4]+54*p[0]^2*p[ 2]^2+144*p[0]^2*p[2]*p[3]+180*p[0]^2*p[2]*p[4]+96*p[0]^2*p[3]^2+240*p[0]^2*p[3] *p[4]+150*p[0]^2*p[4]^2+32*p[0]*p[1]^3+144*p[0]*p[1]^2*p[2]+192*p[0]*p[1]^2*p[3 ]+240*p[0]*p[1]^2*p[4]+216*p[0]*p[1]*p[2]^2+576*p[0]*p[1]*p[2]*p[3]+720*p[0]*p[ 1]*p[2]*p[4]+384*p[0]*p[1]*p[3]^2+960*p[0]*p[1]*p[3]*p[4]+600*p[0]*p[1]*p[4]^2+ 108*p[0]*p[2]^3+432*p[0]*p[2]^2*p[3]+540*p[0]*p[2]^2*p[4]+576*p[0]*p[2]*p[3]^2+ 1440*p[0]*p[2]*p[3]*p[4]+900*p[0]*p[2]*p[4]^2+256*p[0]*p[3]^3+960*p[0]*p[3]^2*p [4]+1200*p[0]*p[3]*p[4]^2+500*p[0]*p[4]^3+16*p[1]^4+96*p[1]^3*p[2]+128*p[1]^3*p [3]+160*p[1]^3*p[4]+216*p[1]^2*p[2]^2+576*p[1]^2*p[2]*p[3]+720*p[1]^2*p[2]*p[4] +384*p[1]^2*p[3]^2+960*p[1]^2*p[3]*p[4]+600*p[1]^2*p[4]^2+216*p[1]*p[2]^3+864*p [1]*p[2]^2*p[3]+1080*p[1]*p[2]^2*p[4]+1152*p[1]*p[2]*p[3]^2+2880*p[1]*p[2]*p[3] *p[4]+1800*p[1]*p[2]*p[4]^2+512*p[1]*p[3]^3+1920*p[1]*p[3]^2*p[4]+2400*p[1]*p[3 ]*p[4]^2+1000*p[1]*p[4]^3+81*p[2]^4+432*p[2]^3*p[3]+540*p[2]^3*p[4]+864*p[2]^2* p[3]^2+2160*p[2]^2*p[3]*p[4]+1350*p[2]^2*p[4]^2+768*p[2]*p[3]^3+2880*p[2]*p[3]^ 2*p[4]+3600*p[2]*p[3]*p[4]^2+1500*p[2]*p[4]^3+256*p[3]^4+1280*p[3]^3*p[4]+2400* p[3]^2*p[4]^2+2000*p[3]*p[4]^3+625*p[4]^4-2*p[0]^3-16*p[0]^2*p[1]-30*p[0]^2*p[2 ]-48*p[0]^2*p[3]-70*p[0]^2*p[4]-40*p[0]*p[1]^2-144*p[0]*p[1]*p[2]-224*p[0]*p[1] *p[3]-320*p[0]*p[1]*p[4]-126*p[0]*p[2]^2-384*p[0]*p[2]*p[3]-540*p[0]*p[2]*p[4]-\ 288*p[0]*p[3]^2-800*p[0]*p[3]*p[4]-550*p[0]*p[4]^2-32*p[1]^3-168*p[1]^2*p[2]-\ 256*p[1]^2*p[3]-360*p[1]^2*p[4]-288*p[1]*p[2]^2-864*p[1]*p[2]*p[3]-1200*p[1]*p[ 2]*p[4]-640*p[1]*p[3]^2-1760*p[1]*p[3]*p[4]-1200*p[1]*p[4]^2-162*p[2]^3-720*p[2 ]^2*p[3]-990*p[2]^2*p[4]-1056*p[2]*p[3]^2-2880*p[2]*p[3]*p[4]-1950*p[2]*p[4]^2-\ 512*p[3]^3-2080*p[3]^2*p[4]-2800*p[3]*p[4]^2-1250*p[4]^3+2*p[0]^2+4*p[0]*p[1]-6 *p[0]*p[2]-32*p[0]*p[3]-80*p[0]*p[4]+12*p[1]^2+48*p[1]*p[2]+64*p[1]*p[3]+60*p[1 ]*p[4]+72*p[2]^2+264*p[2]*p[3]+390*p[2]*p[4]+272*p[3]^2+880*p[3]*p[4]+750*p[4]^ 2-2*p[0]+4*p[1]+24*p[2]+64*p[3]+130*p[4]+1)/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]-1 )^5/((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+10*p[0]*p[4]+4*p[1]^2+12*p[1]* p[2]+16*p[1]*p[3]+20*p[1]*p[4]+9*p[2]^2+24*p[2]*p[3]+30*p[2]*p[4]+16*p[3]^2+40* p[3]*p[4]+25*p[4]^2-p[0]-4*p[1]-9*p[2]-16*p[3]-25*p[4])/(p[0]+2*p[1]+3*p[2]+4*p [3]+5*p[4]-1)^3)^(3/2) The explicit expression for the, 4, -th scaled moment is 6 5 5 5 5 (p[0] + 12 p[0] p[1] + 18 p[0] p[2] + 24 p[0] p[3] + 30 p[0] p[4] 4 2 4 4 + 60 p[0] p[1] + 180 p[0] p[1] p[2] + 240 p[0] p[1] p[3] 4 4 2 4 + 300 p[0] p[1] p[4] + 135 p[0] p[2] + 360 p[0] p[2] p[3] 4 4 2 4 + 450 p[0] p[2] p[4] + 240 p[0] p[3] + 600 p[0] p[3] p[4] 4 2 3 3 3 2 + 375 p[0] p[4] + 160 p[0] p[1] + 720 p[0] p[1] p[2] 3 2 3 2 3 2 + 960 p[0] p[1] p[3] + 1200 p[0] p[1] p[4] + 1080 p[0] p[1] p[2] 3 3 + 2880 p[0] p[1] p[2] p[3] + 3600 p[0] p[1] p[2] p[4] 3 2 3 + 1920 p[0] p[1] p[3] + 4800 p[0] p[1] p[3] p[4] 3 2 3 3 3 2 + 3000 p[0] p[1] p[4] + 540 p[0] p[2] + 2160 p[0] p[2] p[3] 3 2 3 2 + 2700 p[0] p[2] p[4] + 2880 p[0] p[2] p[3] 3 3 2 3 3 + 7200 p[0] p[2] p[3] p[4] + 4500 p[0] p[2] p[4] + 1280 p[0] p[3] 3 2 3 2 3 3 + 4800 p[0] p[3] p[4] + 6000 p[0] p[3] p[4] + 2500 p[0] p[4] 2 4 2 3 2 3 + 240 p[0] p[1] + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] 2 3 2 2 2 + 2400 p[0] p[1] p[4] + 3240 p[0] p[1] p[2] 2 2 2 2 + 8640 p[0] p[1] p[2] p[3] + 10800 p[0] p[1] p[2] p[4] 2 2 2 2 2 + 5760 p[0] p[1] p[3] + 14400 p[0] p[1] p[3] p[4] 2 2 2 2 3 + 9000 p[0] p[1] p[4] + 3240 p[0] p[1] p[2] 2 2 2 2 + 12960 p[0] p[1] p[2] p[3] + 16200 p[0] p[1] p[2] p[4] 2 2 2 + 17280 p[0] p[1] p[2] p[3] + 43200 p[0] p[1] p[2] p[3] p[4] 2 2 2 3 + 27000 p[0] p[1] p[2] p[4] + 7680 p[0] p[1] p[3] 2 2 2 2 + 28800 p[0] p[1] p[3] p[4] + 36000 p[0] p[1] p[3] p[4] 2 3 2 4 2 3 + 15000 p[0] p[1] p[4] + 1215 p[0] p[2] + 6480 p[0] p[2] p[3] 2 3 2 2 2 + 8100 p[0] p[2] p[4] + 12960 p[0] p[2] p[3] 2 2 2 2 2 + 32400 p[0] p[2] p[3] p[4] + 20250 p[0] p[2] p[4] 2 3 2 2 + 11520 p[0] p[2] p[3] + 43200 p[0] p[2] p[3] p[4] 2 2 2 3 2 4 + 54000 p[0] p[2] p[3] p[4] + 22500 p[0] p[2] p[4] + 3840 p[0] p[3] 2 3 2 2 2 + 19200 p[0] p[3] p[4] + 36000 p[0] p[3] p[4] 2 3 2 4 5 + 30000 p[0] p[3] p[4] + 9375 p[0] p[4] + 192 p[0] p[1] 4 4 4 + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] + 2400 p[0] p[1] p[4] 3 2 3 + 4320 p[0] p[1] p[2] + 11520 p[0] p[1] p[2] p[3] 3 3 2 + 14400 p[0] p[1] p[2] p[4] + 7680 p[0] p[1] p[3] 3 3 2 + 19200 p[0] p[1] p[3] p[4] + 12000 p[0] p[1] p[4] 2 3 2 2 + 6480 p[0] p[1] p[2] + 25920 p[0] p[1] p[2] p[3] 2 2 2 2 + 32400 p[0] p[1] p[2] p[4] + 34560 p[0] p[1] p[2] p[3] 2 2 2 + 86400 p[0] p[1] p[2] p[3] p[4] + 54000 p[0] p[1] p[2] p[4] 2 3 2 2 + 15360 p[0] p[1] p[3] + 57600 p[0] p[1] p[3] p[4] 2 2 2 3 + 72000 p[0] p[1] p[3] p[4] + 30000 p[0] p[1] p[4] 4 3 + 4860 p[0] p[1] p[2] + 25920 p[0] p[1] p[2] p[3] 3 2 2 + 32400 p[0] p[1] p[2] p[4] + 51840 p[0] p[1] p[2] p[3] 2 2 2 + 129600 p[0] p[1] p[2] p[3] p[4] + 81000 p[0] p[1] p[2] p[4] 3 2 + 46080 p[0] p[1] p[2] p[3] + 172800 p[0] p[1] p[2] p[3] p[4] 2 3 + 216000 p[0] p[1] p[2] p[3] p[4] + 90000 p[0] p[1] p[2] p[4] 4 3 + 15360 p[0] p[1] p[3] + 76800 p[0] p[1] p[3] p[4] 2 2 3 + 144000 p[0] p[1] p[3] p[4] + 120000 p[0] p[1] p[3] p[4] 4 5 4 + 37500 p[0] p[1] p[4] + 1458 p[0] p[2] + 9720 p[0] p[2] p[3] 4 3 2 + 12150 p[0] p[2] p[4] + 25920 p[0] p[2] p[3] 3 3 2 + 64800 p[0] p[2] p[3] p[4] + 40500 p[0] p[2] p[4] 2 3 2 2 + 34560 p[0] p[2] p[3] + 129600 p[0] p[2] p[3] p[4] 2 2 2 3 + 162000 p[0] p[2] p[3] p[4] + 67500 p[0] p[2] p[4] 4 3 + 23040 p[0] p[2] p[3] + 115200 p[0] p[2] p[3] p[4] 2 2 3 + 216000 p[0] p[2] p[3] p[4] + 180000 p[0] p[2] p[3] p[4] 4 5 4 + 56250 p[0] p[2] p[4] + 6144 p[0] p[3] + 38400 p[0] p[3] p[4] 3 2 2 3 4 + 96000 p[0] p[3] p[4] + 120000 p[0] p[3] p[4] + 75000 p[0] p[3] p[4] 5 6 5 5 + 18750 p[0] p[4] + 64 p[1] + 576 p[1] p[2] + 768 p[1] p[3] 5 4 2 4 + 960 p[1] p[4] + 2160 p[1] p[2] + 5760 p[1] p[2] p[3] 4 4 2 4 + 7200 p[1] p[2] p[4] + 3840 p[1] p[3] + 9600 p[1] p[3] p[4] 4 2 3 3 3 2 + 6000 p[1] p[4] + 4320 p[1] p[2] + 17280 p[1] p[2] p[3] 3 2 3 2 + 21600 p[1] p[2] p[4] + 23040 p[1] p[2] p[3] 3 3 2 3 3 + 57600 p[1] p[2] p[3] p[4] + 36000 p[1] p[2] p[4] + 10240 p[1] p[3] 3 2 3 2 3 3 + 38400 p[1] p[3] p[4] + 48000 p[1] p[3] p[4] + 20000 p[1] p[4] 2 4 2 3 2 3 + 4860 p[1] p[2] + 25920 p[1] p[2] p[3] + 32400 p[1] p[2] p[4] 2 2 2 2 2 + 51840 p[1] p[2] p[3] + 129600 p[1] p[2] p[3] p[4] 2 2 2 2 3 + 81000 p[1] p[2] p[4] + 46080 p[1] p[2] p[3] 2 2 2 2 + 172800 p[1] p[2] p[3] p[4] + 216000 p[1] p[2] p[3] p[4] 2 3 2 4 2 3 + 90000 p[1] p[2] p[4] + 15360 p[1] p[3] + 76800 p[1] p[3] p[4] 2 2 2 2 3 2 4 + 144000 p[1] p[3] p[4] + 120000 p[1] p[3] p[4] + 37500 p[1] p[4] 5 4 4 + 2916 p[1] p[2] + 19440 p[1] p[2] p[3] + 24300 p[1] p[2] p[4] 3 2 3 + 51840 p[1] p[2] p[3] + 129600 p[1] p[2] p[3] p[4] 3 2 2 3 + 81000 p[1] p[2] p[4] + 69120 p[1] p[2] p[3] 2 2 2 2 + 259200 p[1] p[2] p[3] p[4] + 324000 p[1] p[2] p[3] p[4] 2 3 4 + 135000 p[1] p[2] p[4] + 46080 p[1] p[2] p[3] 3 2 2 + 230400 p[1] p[2] p[3] p[4] + 432000 p[1] p[2] p[3] p[4] 3 4 5 + 360000 p[1] p[2] p[3] p[4] + 112500 p[1] p[2] p[4] + 12288 p[1] p[3] 4 3 2 + 76800 p[1] p[3] p[4] + 192000 p[1] p[3] p[4] 2 3 4 5 + 240000 p[1] p[3] p[4] + 150000 p[1] p[3] p[4] + 37500 p[1] p[4] 6 5 5 4 2 + 729 p[2] + 5832 p[2] p[3] + 7290 p[2] p[4] + 19440 p[2] p[3] 4 4 2 3 3 + 48600 p[2] p[3] p[4] + 30375 p[2] p[4] + 34560 p[2] p[3] 3 2 3 2 3 3 + 129600 p[2] p[3] p[4] + 162000 p[2] p[3] p[4] + 67500 p[2] p[4] 2 4 2 3 2 2 2 + 34560 p[2] p[3] + 172800 p[2] p[3] p[4] + 324000 p[2] p[3] p[4] 2 3 2 4 5 + 270000 p[2] p[3] p[4] + 84375 p[2] p[4] + 18432 p[2] p[3] 4 3 2 + 115200 p[2] p[3] p[4] + 288000 p[2] p[3] p[4] 2 3 4 5 + 360000 p[2] p[3] p[4] + 225000 p[2] p[3] p[4] + 56250 p[2] p[4] 6 5 4 2 3 3 + 4096 p[3] + 30720 p[3] p[4] + 96000 p[3] p[4] + 160000 p[3] p[4] 2 4 5 6 4 + 150000 p[3] p[4] + 75000 p[3] p[4] + 15625 p[4] - 6 p[0] p[1] 4 4 4 3 2 - 18 p[0] p[2] - 36 p[0] p[3] - 60 p[0] p[4] - 48 p[0] p[1] 3 3 3 - 216 p[0] p[1] p[2] - 384 p[0] p[1] p[3] - 600 p[0] p[1] p[4] 3 2 3 3 - 216 p[0] p[2] - 720 p[0] p[2] p[3] - 1080 p[0] p[2] p[4] 3 2 3 3 2 - 576 p[0] p[3] - 1680 p[0] p[3] p[4] - 1200 p[0] p[4] 2 3 2 2 2 2 - 144 p[0] p[1] - 864 p[0] p[1] p[2] - 1440 p[0] p[1] p[3] 2 2 2 2 - 2160 p[0] p[1] p[4] - 1620 p[0] p[1] p[2] 2 2 - 5184 p[0] p[1] p[2] p[3] - 7560 p[0] p[1] p[2] p[4] 2 2 2 - 4032 p[0] p[1] p[3] - 11520 p[0] p[1] p[3] p[4] 2 2 2 3 2 2 - 8100 p[0] p[1] p[4] - 972 p[0] p[2] - 4536 p[0] p[2] p[3] 2 2 2 2 - 6480 p[0] p[2] p[4] - 6912 p[0] p[2] p[3] 2 2 2 2 3 - 19440 p[0] p[2] p[3] p[4] - 13500 p[0] p[2] p[4] - 3456 p[0] p[3] 2 2 2 2 2 3 - 14400 p[0] p[3] p[4] - 19800 p[0] p[3] p[4] - 9000 p[0] p[4] 4 3 3 - 192 p[0] p[1] - 1440 p[0] p[1] p[2] - 2304 p[0] p[1] p[3] 3 2 2 - 3360 p[0] p[1] p[4] - 3888 p[0] p[1] p[2] 2 2 - 12096 p[0] p[1] p[2] p[3] - 17280 p[0] p[1] p[2] p[4] 2 2 2 - 9216 p[0] p[1] p[3] - 25920 p[0] p[1] p[3] p[4] 2 2 3 - 18000 p[0] p[1] p[4] - 4536 p[0] p[1] p[2] 2 2 - 20736 p[0] p[1] p[2] p[3] - 29160 p[0] p[1] p[2] p[4] 2 - 31104 p[0] p[1] p[2] p[3] - 86400 p[0] p[1] p[2] p[3] p[4] 2 3 - 59400 p[0] p[1] p[2] p[4] - 15360 p[0] p[1] p[3] 2 2 - 63360 p[0] p[1] p[3] p[4] - 86400 p[0] p[1] p[3] p[4] 3 4 3 - 39000 p[0] p[1] p[4] - 1944 p[0] p[2] - 11664 p[0] p[2] p[3] 3 2 2 - 16200 p[0] p[2] p[4] - 25920 p[0] p[2] p[3] 2 2 2 - 71280 p[0] p[2] p[3] p[4] - 48600 p[0] p[2] p[4] 3 2 - 25344 p[0] p[2] p[3] - 103680 p[0] p[2] p[3] p[4] 2 3 4 - 140400 p[0] p[2] p[3] p[4] - 63000 p[0] p[2] p[4] - 9216 p[0] p[3] 3 2 2 3 - 49920 p[0] p[3] p[4] - 100800 p[0] p[3] p[4] - 90000 p[0] p[3] p[4] 4 5 4 4 - 30000 p[0] p[4] - 96 p[1] - 864 p[1] p[2] - 1344 p[1] p[3] 4 3 2 3 - 1920 p[1] p[4] - 3024 p[1] p[2] - 9216 p[1] p[2] p[3] 3 3 2 3 - 12960 p[1] p[2] p[4] - 6912 p[1] p[3] - 19200 p[1] p[3] p[4] 3 2 2 3 2 2 - 13200 p[1] p[4] - 5184 p[1] p[2] - 23328 p[1] p[2] p[3] 2 2 2 2 - 32400 p[1] p[2] p[4] - 34560 p[1] p[2] p[3] 2 2 2 2 3 - 95040 p[1] p[2] p[3] p[4] - 64800 p[1] p[2] p[4] - 16896 p[1] p[3] 2 2 2 2 2 3 - 69120 p[1] p[3] p[4] - 93600 p[1] p[3] p[4] - 42000 p[1] p[4] 4 3 3 - 4374 p[1] p[2] - 25920 p[1] p[2] p[3] - 35640 p[1] p[2] p[4] 2 2 2 - 57024 p[1] p[2] p[3] - 155520 p[1] p[2] p[3] p[4] 2 2 3 - 105300 p[1] p[2] p[4] - 55296 p[1] p[2] p[3] 2 2 - 224640 p[1] p[2] p[3] p[4] - 302400 p[1] p[2] p[3] p[4] 3 4 3 - 135000 p[1] p[2] p[4] - 19968 p[1] p[3] - 107520 p[1] p[3] p[4] 2 2 3 4 - 216000 p[1] p[3] p[4] - 192000 p[1] p[3] p[4] - 63750 p[1] p[4] 5 4 4 3 2 - 1458 p[2] - 10692 p[2] p[3] - 14580 p[2] p[4] - 31104 p[2] p[3] 3 3 2 2 3 - 84240 p[2] p[3] p[4] - 56700 p[2] p[4] - 44928 p[2] p[3] 2 2 2 2 2 3 - 181440 p[2] p[3] p[4] - 243000 p[2] p[3] p[4] - 108000 p[2] p[4] 4 3 2 2 - 32256 p[2] p[3] - 172800 p[2] p[3] p[4] - 345600 p[2] p[3] p[4] 3 4 5 - 306000 p[2] p[3] p[4] - 101250 p[2] p[4] - 9216 p[3] 4 3 2 2 3 - 61440 p[3] p[4] - 163200 p[3] p[4] - 216000 p[3] p[4] 4 5 4 3 3 - 142500 p[3] p[4] - 37500 p[4] - p[0] - 36 p[0] p[1] - 114 p[0] p[2] 3 3 2 2 2 - 256 p[0] p[3] - 480 p[0] p[4] - 156 p[0] p[1] - 720 p[0] p[1] p[2] 2 2 2 2 - 1440 p[0] p[1] p[3] - 2580 p[0] p[1] p[4] - 648 p[0] p[2] 2 2 2 2 - 2232 p[0] p[2] p[3] - 3690 p[0] p[2] p[4] - 1680 p[0] p[3] 2 2 2 3 - 5040 p[0] p[3] p[4] - 3450 p[0] p[4] - 224 p[0] p[1] 2 2 2 - 1296 p[0] p[1] p[2] - 2400 p[0] p[1] p[3] - 4200 p[0] p[1] p[4] 2 - 2052 p[0] p[1] p[2] - 6624 p[0] p[1] p[2] p[3] 2 - 10800 p[0] p[1] p[2] p[4] - 4608 p[0] p[1] p[3] 2 3 - 13440 p[0] p[1] p[3] p[4] - 8700 p[0] p[1] p[4] - 918 p[0] p[2] 2 2 2 - 3888 p[0] p[2] p[3] - 5940 p[0] p[2] p[4] - 4608 p[0] p[2] p[3] 2 3 - 12240 p[0] p[2] p[3] p[4] - 6750 p[0] p[2] p[4] - 1408 p[0] p[3] 2 2 3 4 - 4320 p[0] p[3] p[4] - 2400 p[0] p[3] p[4] + 1000 p[0] p[4] - 96 p[1] 3 3 3 2 2 - 624 p[1] p[2] - 1088 p[1] p[3] - 1920 p[1] p[4] - 1188 p[1] p[2] 2 2 2 2 - 3456 p[1] p[2] p[3] - 5760 p[1] p[2] p[4] - 1920 p[1] p[3] 2 2 2 3 - 5280 p[1] p[3] p[4] - 2700 p[1] p[4] - 648 p[1] p[2] 2 2 2 - 1728 p[1] p[2] p[3] - 2700 p[1] p[2] p[4] + 576 p[1] p[2] p[3] 2 3 + 4320 p[1] p[2] p[3] p[4] + 7200 p[1] p[2] p[4] + 2304 p[1] p[3] 2 2 3 + 13440 p[1] p[3] p[4] + 26400 p[1] p[3] p[4] + 16500 p[1] p[4] 4 3 3 2 2 + 81 p[2] + 1512 p[2] p[3] + 2430 p[2] p[4] + 6480 p[2] p[3] 2 2 2 3 + 21600 p[2] p[3] p[4] + 18900 p[2] p[4] + 9984 p[2] p[3] 2 2 3 + 48960 p[2] p[3] p[4] + 81000 p[2] p[3] p[4] + 44250 p[2] p[4] 4 3 2 2 3 + 5120 p[3] + 32640 p[3] p[4] + 78000 p[3] p[4] + 82000 p[3] p[4] 4 3 2 2 2 + 31875 p[4] - 7 p[0] - 6 p[0] p[1] + 27 p[0] p[2] + 36 p[0] p[3] 2 2 - 45 p[0] p[4] + 180 p[0] p[1] + 1164 p[0] p[1] p[2] 2 + 2352 p[0] p[1] p[3] + 3780 p[0] p[1] p[4] + 1791 p[0] p[2] 2 + 7416 p[0] p[2] p[3] + 12630 p[0] p[2] p[4] + 7824 p[0] p[3] 2 3 2 + 27240 p[0] p[3] p[4] + 24075 p[0] p[4] + 208 p[1] + 1500 p[1] p[2] 2 2 2 + 2880 p[1] p[3] + 4740 p[1] p[4] + 3186 p[1] p[2] 2 + 11856 p[1] p[2] p[3] + 19740 p[1] p[2] p[4] + 10944 p[1] p[3] 2 3 2 + 36720 p[1] p[3] p[4] + 31050 p[1] p[4] + 1701 p[2] + 8244 p[2] p[3] 2 2 + 13095 p[2] p[4] + 11952 p[2] p[3] + 36120 p[2] p[3] p[4] 2 3 2 2 + 26475 p[2] p[4] + 4352 p[3] + 16080 p[3] p[4] + 15300 p[3] p[4] 3 2 + 625 p[4] + 12 p[0] + 84 p[0] p[1] + 270 p[0] p[2] + 720 p[0] p[3] 2 + 1620 p[0] p[4] - 36 p[1] - 408 p[1] p[2] - 768 p[1] p[3] 2 - 900 p[1] p[4] - 1062 p[2] - 4824 p[2] p[3] - 8310 p[2] p[4] 2 2 - 5808 p[3] - 21120 p[3] p[4] - 19800 p[4] - 5 p[0] - 48 p[1] - 183 p[2] / 2 - 488 p[3] - 1065 p[4]) / ((p[0] + 4 p[0] p[1] + 6 p[0] p[2] / 2 + 8 p[0] p[3] + 10 p[0] p[4] + 4 p[1] + 12 p[1] p[2] + 16 p[1] p[3] 2 2 + 20 p[1] p[4] + 9 p[2] + 24 p[2] p[3] + 30 p[2] p[4] + 16 p[3] 2 2 + 40 p[3] p[4] + 25 p[4] - p[0] - 4 p[1] - 9 p[2] - 16 p[3] - 25 p[4]) (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] - 1)) and in Maple notation (p[0]^6+12*p[0]^5*p[1]+18*p[0]^5*p[2]+24*p[0]^5*p[3]+30*p[0]^5*p[4]+60*p[0]^4*p [1]^2+180*p[0]^4*p[1]*p[2]+240*p[0]^4*p[1]*p[3]+300*p[0]^4*p[1]*p[4]+135*p[0]^4 *p[2]^2+360*p[0]^4*p[2]*p[3]+450*p[0]^4*p[2]*p[4]+240*p[0]^4*p[3]^2+600*p[0]^4* p[3]*p[4]+375*p[0]^4*p[4]^2+160*p[0]^3*p[1]^3+720*p[0]^3*p[1]^2*p[2]+960*p[0]^3 *p[1]^2*p[3]+1200*p[0]^3*p[1]^2*p[4]+1080*p[0]^3*p[1]*p[2]^2+2880*p[0]^3*p[1]*p [2]*p[3]+3600*p[0]^3*p[1]*p[2]*p[4]+1920*p[0]^3*p[1]*p[3]^2+4800*p[0]^3*p[1]*p[ 3]*p[4]+3000*p[0]^3*p[1]*p[4]^2+540*p[0]^3*p[2]^3+2160*p[0]^3*p[2]^2*p[3]+2700* p[0]^3*p[2]^2*p[4]+2880*p[0]^3*p[2]*p[3]^2+7200*p[0]^3*p[2]*p[3]*p[4]+4500*p[0] ^3*p[2]*p[4]^2+1280*p[0]^3*p[3]^3+4800*p[0]^3*p[3]^2*p[4]+6000*p[0]^3*p[3]*p[4] ^2+2500*p[0]^3*p[4]^3+240*p[0]^2*p[1]^4+1440*p[0]^2*p[1]^3*p[2]+1920*p[0]^2*p[1 ]^3*p[3]+2400*p[0]^2*p[1]^3*p[4]+3240*p[0]^2*p[1]^2*p[2]^2+8640*p[0]^2*p[1]^2*p [2]*p[3]+10800*p[0]^2*p[1]^2*p[2]*p[4]+5760*p[0]^2*p[1]^2*p[3]^2+14400*p[0]^2*p [1]^2*p[3]*p[4]+9000*p[0]^2*p[1]^2*p[4]^2+3240*p[0]^2*p[1]*p[2]^3+12960*p[0]^2* p[1]*p[2]^2*p[3]+16200*p[0]^2*p[1]*p[2]^2*p[4]+17280*p[0]^2*p[1]*p[2]*p[3]^2+ 43200*p[0]^2*p[1]*p[2]*p[3]*p[4]+27000*p[0]^2*p[1]*p[2]*p[4]^2+7680*p[0]^2*p[1] *p[3]^3+28800*p[0]^2*p[1]*p[3]^2*p[4]+36000*p[0]^2*p[1]*p[3]*p[4]^2+15000*p[0]^ 2*p[1]*p[4]^3+1215*p[0]^2*p[2]^4+6480*p[0]^2*p[2]^3*p[3]+8100*p[0]^2*p[2]^3*p[4 ]+12960*p[0]^2*p[2]^2*p[3]^2+32400*p[0]^2*p[2]^2*p[3]*p[4]+20250*p[0]^2*p[2]^2* p[4]^2+11520*p[0]^2*p[2]*p[3]^3+43200*p[0]^2*p[2]*p[3]^2*p[4]+54000*p[0]^2*p[2] *p[3]*p[4]^2+22500*p[0]^2*p[2]*p[4]^3+3840*p[0]^2*p[3]^4+19200*p[0]^2*p[3]^3*p[ 4]+36000*p[0]^2*p[3]^2*p[4]^2+30000*p[0]^2*p[3]*p[4]^3+9375*p[0]^2*p[4]^4+192*p [0]*p[1]^5+1440*p[0]*p[1]^4*p[2]+1920*p[0]*p[1]^4*p[3]+2400*p[0]*p[1]^4*p[4]+ 4320*p[0]*p[1]^3*p[2]^2+11520*p[0]*p[1]^3*p[2]*p[3]+14400*p[0]*p[1]^3*p[2]*p[4] +7680*p[0]*p[1]^3*p[3]^2+19200*p[0]*p[1]^3*p[3]*p[4]+12000*p[0]*p[1]^3*p[4]^2+ 6480*p[0]*p[1]^2*p[2]^3+25920*p[0]*p[1]^2*p[2]^2*p[3]+32400*p[0]*p[1]^2*p[2]^2* p[4]+34560*p[0]*p[1]^2*p[2]*p[3]^2+86400*p[0]*p[1]^2*p[2]*p[3]*p[4]+54000*p[0]* p[1]^2*p[2]*p[4]^2+15360*p[0]*p[1]^2*p[3]^3+57600*p[0]*p[1]^2*p[3]^2*p[4]+72000 *p[0]*p[1]^2*p[3]*p[4]^2+30000*p[0]*p[1]^2*p[4]^3+4860*p[0]*p[1]*p[2]^4+25920*p [0]*p[1]*p[2]^3*p[3]+32400*p[0]*p[1]*p[2]^3*p[4]+51840*p[0]*p[1]*p[2]^2*p[3]^2+ 129600*p[0]*p[1]*p[2]^2*p[3]*p[4]+81000*p[0]*p[1]*p[2]^2*p[4]^2+46080*p[0]*p[1] *p[2]*p[3]^3+172800*p[0]*p[1]*p[2]*p[3]^2*p[4]+216000*p[0]*p[1]*p[2]*p[3]*p[4]^ 2+90000*p[0]*p[1]*p[2]*p[4]^3+15360*p[0]*p[1]*p[3]^4+76800*p[0]*p[1]*p[3]^3*p[4 ]+144000*p[0]*p[1]*p[3]^2*p[4]^2+120000*p[0]*p[1]*p[3]*p[4]^3+37500*p[0]*p[1]*p [4]^4+1458*p[0]*p[2]^5+9720*p[0]*p[2]^4*p[3]+12150*p[0]*p[2]^4*p[4]+25920*p[0]* p[2]^3*p[3]^2+64800*p[0]*p[2]^3*p[3]*p[4]+40500*p[0]*p[2]^3*p[4]^2+34560*p[0]*p [2]^2*p[3]^3+129600*p[0]*p[2]^2*p[3]^2*p[4]+162000*p[0]*p[2]^2*p[3]*p[4]^2+ 67500*p[0]*p[2]^2*p[4]^3+23040*p[0]*p[2]*p[3]^4+115200*p[0]*p[2]*p[3]^3*p[4]+ 216000*p[0]*p[2]*p[3]^2*p[4]^2+180000*p[0]*p[2]*p[3]*p[4]^3+56250*p[0]*p[2]*p[4 ]^4+6144*p[0]*p[3]^5+38400*p[0]*p[3]^4*p[4]+96000*p[0]*p[3]^3*p[4]^2+120000*p[0 ]*p[3]^2*p[4]^3+75000*p[0]*p[3]*p[4]^4+18750*p[0]*p[4]^5+64*p[1]^6+576*p[1]^5*p [2]+768*p[1]^5*p[3]+960*p[1]^5*p[4]+2160*p[1]^4*p[2]^2+5760*p[1]^4*p[2]*p[3]+ 7200*p[1]^4*p[2]*p[4]+3840*p[1]^4*p[3]^2+9600*p[1]^4*p[3]*p[4]+6000*p[1]^4*p[4] ^2+4320*p[1]^3*p[2]^3+17280*p[1]^3*p[2]^2*p[3]+21600*p[1]^3*p[2]^2*p[4]+23040*p [1]^3*p[2]*p[3]^2+57600*p[1]^3*p[2]*p[3]*p[4]+36000*p[1]^3*p[2]*p[4]^2+10240*p[ 1]^3*p[3]^3+38400*p[1]^3*p[3]^2*p[4]+48000*p[1]^3*p[3]*p[4]^2+20000*p[1]^3*p[4] ^3+4860*p[1]^2*p[2]^4+25920*p[1]^2*p[2]^3*p[3]+32400*p[1]^2*p[2]^3*p[4]+51840*p [1]^2*p[2]^2*p[3]^2+129600*p[1]^2*p[2]^2*p[3]*p[4]+81000*p[1]^2*p[2]^2*p[4]^2+ 46080*p[1]^2*p[2]*p[3]^3+172800*p[1]^2*p[2]*p[3]^2*p[4]+216000*p[1]^2*p[2]*p[3] *p[4]^2+90000*p[1]^2*p[2]*p[4]^3+15360*p[1]^2*p[3]^4+76800*p[1]^2*p[3]^3*p[4]+ 144000*p[1]^2*p[3]^2*p[4]^2+120000*p[1]^2*p[3]*p[4]^3+37500*p[1]^2*p[4]^4+2916* p[1]*p[2]^5+19440*p[1]*p[2]^4*p[3]+24300*p[1]*p[2]^4*p[4]+51840*p[1]*p[2]^3*p[3 ]^2+129600*p[1]*p[2]^3*p[3]*p[4]+81000*p[1]*p[2]^3*p[4]^2+69120*p[1]*p[2]^2*p[3 ]^3+259200*p[1]*p[2]^2*p[3]^2*p[4]+324000*p[1]*p[2]^2*p[3]*p[4]^2+135000*p[1]*p [2]^2*p[4]^3+46080*p[1]*p[2]*p[3]^4+230400*p[1]*p[2]*p[3]^3*p[4]+432000*p[1]*p[ 2]*p[3]^2*p[4]^2+360000*p[1]*p[2]*p[3]*p[4]^3+112500*p[1]*p[2]*p[4]^4+12288*p[1 ]*p[3]^5+76800*p[1]*p[3]^4*p[4]+192000*p[1]*p[3]^3*p[4]^2+240000*p[1]*p[3]^2*p[ 4]^3+150000*p[1]*p[3]*p[4]^4+37500*p[1]*p[4]^5+729*p[2]^6+5832*p[2]^5*p[3]+7290 *p[2]^5*p[4]+19440*p[2]^4*p[3]^2+48600*p[2]^4*p[3]*p[4]+30375*p[2]^4*p[4]^2+ 34560*p[2]^3*p[3]^3+129600*p[2]^3*p[3]^2*p[4]+162000*p[2]^3*p[3]*p[4]^2+67500*p [2]^3*p[4]^3+34560*p[2]^2*p[3]^4+172800*p[2]^2*p[3]^3*p[4]+324000*p[2]^2*p[3]^2 *p[4]^2+270000*p[2]^2*p[3]*p[4]^3+84375*p[2]^2*p[4]^4+18432*p[2]*p[3]^5+115200* p[2]*p[3]^4*p[4]+288000*p[2]*p[3]^3*p[4]^2+360000*p[2]*p[3]^2*p[4]^3+225000*p[2 ]*p[3]*p[4]^4+56250*p[2]*p[4]^5+4096*p[3]^6+30720*p[3]^5*p[4]+96000*p[3]^4*p[4] ^2+160000*p[3]^3*p[4]^3+150000*p[3]^2*p[4]^4+75000*p[3]*p[4]^5+15625*p[4]^6-6*p [0]^4*p[1]-18*p[0]^4*p[2]-36*p[0]^4*p[3]-60*p[0]^4*p[4]-48*p[0]^3*p[1]^2-216*p[ 0]^3*p[1]*p[2]-384*p[0]^3*p[1]*p[3]-600*p[0]^3*p[1]*p[4]-216*p[0]^3*p[2]^2-720* p[0]^3*p[2]*p[3]-1080*p[0]^3*p[2]*p[4]-576*p[0]^3*p[3]^2-1680*p[0]^3*p[3]*p[4]-\ 1200*p[0]^3*p[4]^2-144*p[0]^2*p[1]^3-864*p[0]^2*p[1]^2*p[2]-1440*p[0]^2*p[1]^2* p[3]-2160*p[0]^2*p[1]^2*p[4]-1620*p[0]^2*p[1]*p[2]^2-5184*p[0]^2*p[1]*p[2]*p[3] -7560*p[0]^2*p[1]*p[2]*p[4]-4032*p[0]^2*p[1]*p[3]^2-11520*p[0]^2*p[1]*p[3]*p[4] -8100*p[0]^2*p[1]*p[4]^2-972*p[0]^2*p[2]^3-4536*p[0]^2*p[2]^2*p[3]-6480*p[0]^2* p[2]^2*p[4]-6912*p[0]^2*p[2]*p[3]^2-19440*p[0]^2*p[2]*p[3]*p[4]-13500*p[0]^2*p[ 2]*p[4]^2-3456*p[0]^2*p[3]^3-14400*p[0]^2*p[3]^2*p[4]-19800*p[0]^2*p[3]*p[4]^2-\ 9000*p[0]^2*p[4]^3-192*p[0]*p[1]^4-1440*p[0]*p[1]^3*p[2]-2304*p[0]*p[1]^3*p[3]-\ 3360*p[0]*p[1]^3*p[4]-3888*p[0]*p[1]^2*p[2]^2-12096*p[0]*p[1]^2*p[2]*p[3]-17280 *p[0]*p[1]^2*p[2]*p[4]-9216*p[0]*p[1]^2*p[3]^2-25920*p[0]*p[1]^2*p[3]*p[4]-\ 18000*p[0]*p[1]^2*p[4]^2-4536*p[0]*p[1]*p[2]^3-20736*p[0]*p[1]*p[2]^2*p[3]-\ 29160*p[0]*p[1]*p[2]^2*p[4]-31104*p[0]*p[1]*p[2]*p[3]^2-86400*p[0]*p[1]*p[2]*p[ 3]*p[4]-59400*p[0]*p[1]*p[2]*p[4]^2-15360*p[0]*p[1]*p[3]^3-63360*p[0]*p[1]*p[3] ^2*p[4]-86400*p[0]*p[1]*p[3]*p[4]^2-39000*p[0]*p[1]*p[4]^3-1944*p[0]*p[2]^4-\ 11664*p[0]*p[2]^3*p[3]-16200*p[0]*p[2]^3*p[4]-25920*p[0]*p[2]^2*p[3]^2-71280*p[ 0]*p[2]^2*p[3]*p[4]-48600*p[0]*p[2]^2*p[4]^2-25344*p[0]*p[2]*p[3]^3-103680*p[0] *p[2]*p[3]^2*p[4]-140400*p[0]*p[2]*p[3]*p[4]^2-63000*p[0]*p[2]*p[4]^3-9216*p[0] *p[3]^4-49920*p[0]*p[3]^3*p[4]-100800*p[0]*p[3]^2*p[4]^2-90000*p[0]*p[3]*p[4]^3 -30000*p[0]*p[4]^4-96*p[1]^5-864*p[1]^4*p[2]-1344*p[1]^4*p[3]-1920*p[1]^4*p[4]-\ 3024*p[1]^3*p[2]^2-9216*p[1]^3*p[2]*p[3]-12960*p[1]^3*p[2]*p[4]-6912*p[1]^3*p[3 ]^2-19200*p[1]^3*p[3]*p[4]-13200*p[1]^3*p[4]^2-5184*p[1]^2*p[2]^3-23328*p[1]^2* p[2]^2*p[3]-32400*p[1]^2*p[2]^2*p[4]-34560*p[1]^2*p[2]*p[3]^2-95040*p[1]^2*p[2] *p[3]*p[4]-64800*p[1]^2*p[2]*p[4]^2-16896*p[1]^2*p[3]^3-69120*p[1]^2*p[3]^2*p[4 ]-93600*p[1]^2*p[3]*p[4]^2-42000*p[1]^2*p[4]^3-4374*p[1]*p[2]^4-25920*p[1]*p[2] ^3*p[3]-35640*p[1]*p[2]^3*p[4]-57024*p[1]*p[2]^2*p[3]^2-155520*p[1]*p[2]^2*p[3] *p[4]-105300*p[1]*p[2]^2*p[4]^2-55296*p[1]*p[2]*p[3]^3-224640*p[1]*p[2]*p[3]^2* p[4]-302400*p[1]*p[2]*p[3]*p[4]^2-135000*p[1]*p[2]*p[4]^3-19968*p[1]*p[3]^4-\ 107520*p[1]*p[3]^3*p[4]-216000*p[1]*p[3]^2*p[4]^2-192000*p[1]*p[3]*p[4]^3-63750 *p[1]*p[4]^4-1458*p[2]^5-10692*p[2]^4*p[3]-14580*p[2]^4*p[4]-31104*p[2]^3*p[3]^ 2-84240*p[2]^3*p[3]*p[4]-56700*p[2]^3*p[4]^2-44928*p[2]^2*p[3]^3-181440*p[2]^2* p[3]^2*p[4]-243000*p[2]^2*p[3]*p[4]^2-108000*p[2]^2*p[4]^3-32256*p[2]*p[3]^4-\ 172800*p[2]*p[3]^3*p[4]-345600*p[2]*p[3]^2*p[4]^2-306000*p[2]*p[3]*p[4]^3-\ 101250*p[2]*p[4]^4-9216*p[3]^5-61440*p[3]^4*p[4]-163200*p[3]^3*p[4]^2-216000*p[ 3]^2*p[4]^3-142500*p[3]*p[4]^4-37500*p[4]^5-p[0]^4-36*p[0]^3*p[1]-114*p[0]^3*p[ 2]-256*p[0]^3*p[3]-480*p[0]^3*p[4]-156*p[0]^2*p[1]^2-720*p[0]^2*p[1]*p[2]-1440* p[0]^2*p[1]*p[3]-2580*p[0]^2*p[1]*p[4]-648*p[0]^2*p[2]^2-2232*p[0]^2*p[2]*p[3]-\ 3690*p[0]^2*p[2]*p[4]-1680*p[0]^2*p[3]^2-5040*p[0]^2*p[3]*p[4]-3450*p[0]^2*p[4] ^2-224*p[0]*p[1]^3-1296*p[0]*p[1]^2*p[2]-2400*p[0]*p[1]^2*p[3]-4200*p[0]*p[1]^2 *p[4]-2052*p[0]*p[1]*p[2]^2-6624*p[0]*p[1]*p[2]*p[3]-10800*p[0]*p[1]*p[2]*p[4]-\ 4608*p[0]*p[1]*p[3]^2-13440*p[0]*p[1]*p[3]*p[4]-8700*p[0]*p[1]*p[4]^2-918*p[0]* p[2]^3-3888*p[0]*p[2]^2*p[3]-5940*p[0]*p[2]^2*p[4]-4608*p[0]*p[2]*p[3]^2-12240* p[0]*p[2]*p[3]*p[4]-6750*p[0]*p[2]*p[4]^2-1408*p[0]*p[3]^3-4320*p[0]*p[3]^2*p[4 ]-2400*p[0]*p[3]*p[4]^2+1000*p[0]*p[4]^3-96*p[1]^4-624*p[1]^3*p[2]-1088*p[1]^3* p[3]-1920*p[1]^3*p[4]-1188*p[1]^2*p[2]^2-3456*p[1]^2*p[2]*p[3]-5760*p[1]^2*p[2] *p[4]-1920*p[1]^2*p[3]^2-5280*p[1]^2*p[3]*p[4]-2700*p[1]^2*p[4]^2-648*p[1]*p[2] ^3-1728*p[1]*p[2]^2*p[3]-2700*p[1]*p[2]^2*p[4]+576*p[1]*p[2]*p[3]^2+4320*p[1]*p [2]*p[3]*p[4]+7200*p[1]*p[2]*p[4]^2+2304*p[1]*p[3]^3+13440*p[1]*p[3]^2*p[4]+ 26400*p[1]*p[3]*p[4]^2+16500*p[1]*p[4]^3+81*p[2]^4+1512*p[2]^3*p[3]+2430*p[2]^3 *p[4]+6480*p[2]^2*p[3]^2+21600*p[2]^2*p[3]*p[4]+18900*p[2]^2*p[4]^2+9984*p[2]*p [3]^3+48960*p[2]*p[3]^2*p[4]+81000*p[2]*p[3]*p[4]^2+44250*p[2]*p[4]^3+5120*p[3] ^4+32640*p[3]^3*p[4]+78000*p[3]^2*p[4]^2+82000*p[3]*p[4]^3+31875*p[4]^4-7*p[0]^ 3-6*p[0]^2*p[1]+27*p[0]^2*p[2]+36*p[0]^2*p[3]-45*p[0]^2*p[4]+180*p[0]*p[1]^2+ 1164*p[0]*p[1]*p[2]+2352*p[0]*p[1]*p[3]+3780*p[0]*p[1]*p[4]+1791*p[0]*p[2]^2+ 7416*p[0]*p[2]*p[3]+12630*p[0]*p[2]*p[4]+7824*p[0]*p[3]^2+27240*p[0]*p[3]*p[4]+ 24075*p[0]*p[4]^2+208*p[1]^3+1500*p[1]^2*p[2]+2880*p[1]^2*p[3]+4740*p[1]^2*p[4] +3186*p[1]*p[2]^2+11856*p[1]*p[2]*p[3]+19740*p[1]*p[2]*p[4]+10944*p[1]*p[3]^2+ 36720*p[1]*p[3]*p[4]+31050*p[1]*p[4]^2+1701*p[2]^3+8244*p[2]^2*p[3]+13095*p[2]^ 2*p[4]+11952*p[2]*p[3]^2+36120*p[2]*p[3]*p[4]+26475*p[2]*p[4]^2+4352*p[3]^3+ 16080*p[3]^2*p[4]+15300*p[3]*p[4]^2+625*p[4]^3+12*p[0]^2+84*p[0]*p[1]+270*p[0]* p[2]+720*p[0]*p[3]+1620*p[0]*p[4]-36*p[1]^2-408*p[1]*p[2]-768*p[1]*p[3]-900*p[1 ]*p[4]-1062*p[2]^2-4824*p[2]*p[3]-8310*p[2]*p[4]-5808*p[3]^2-21120*p[3]*p[4]-\ 19800*p[4]^2-5*p[0]-48*p[1]-183*p[2]-488*p[3]-1065*p[4])/(p[0]^2+4*p[0]*p[1]+6* p[0]*p[2]+8*p[0]*p[3]+10*p[0]*p[4]+4*p[1]^2+12*p[1]*p[2]+16*p[1]*p[3]+20*p[1]*p [4]+9*p[2]^2+24*p[2]*p[3]+30*p[2]*p[4]+16*p[3]^2+40*p[3]*p[4]+25*p[4]^2-p[0]-4* p[1]-9*p[2]-16*p[3]-25*p[4])^2/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]-1) This took, 0.212, seconds. The algebraic equation satisfied by the Probability Generating Function of L\ ife in a Casion where the stakes are from, -1, to , 5 By Shalosh B. Ekhad Theorem: Let a(n) be probability that you exit after EXACTLY n rounds in a \ casino where in each turn The probability that you win, 1, dollars is, p[1] The probability that you win, 2, dollars is, p[2] The probability that you win, 3, dollars is, p[3] The probability that you win, 4, dollars is, p[4] The probability that you win, 5, dollars is, p[5] The probability that you lose, -1, dollars is, p[-1] and the probability that you neither win nor lose is, p[0] Of course the probabilities have to add-up to 1, in other wrods p[-1] + p[0] + p[1] + p[2] + p[3] + p[4] + p[5] = 1 We also assume that the Casino stays in business, in other words, the expect\ ed gain in one round is negative: -p[-1] + p[1] + 2 p[2] + 3 p[3] + 4 p[4] + 5 p[5] < 1 Let X(t) be the ordinary generating function of that sequence, in other word\ s infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 X(t) satisifies the algebraic equation 6 5 4 3 2 p[5] t X(t) + X(t) p[4] t + X(t) p[3] t + X(t) p[2] t + X(t) p[1] t + (t p[0] - 1) X(t) + p[-1] t = 0 and in Maple notation p[5]*t*X(t)^6+X(t)^5*p[4]*t+X(t)^4*p[3]*t+X(t)^3*p[2]*t+X(t)^2*p[1]*t+(t*p[0]-1 )*X(t)+p[-1]*t = 0 Or more usefully (for computing many terms) 6 5 4 3 2 X(t) = p[5] t X(t) + X(t) p[4] t + X(t) p[3] t + X(t) p[2] t + X(t) p[1] t + p[0] t X(t) + p[-1] t and in Maple notation X(t) = p[5]*t*X(t)^6+X(t)^5*p[4]*t+X(t)^4*p[3]*t+X(t)^3*p[2]*t+X(t)^2*p[1]*t+p[ 0]*t*X(t)+p[-1]*t The explicit expression for the expected duration in the casino is 1 - ----------------------------------------------------- p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] + 6 p[5] - 1 and in Maple notation -1/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]+6*p[5]-1) The explicit expression for the standard-deviation is / 2 |(p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] + 10 p[0] p[4] + 12 p[0] p[5] \ 2 + 4 p[1] + 12 p[1] p[2] + 16 p[1] p[3] + 20 p[1] p[4] + 24 p[1] p[5] 2 2 + 9 p[2] + 24 p[2] p[3] + 30 p[2] p[4] + 36 p[2] p[5] + 16 p[3] 2 2 + 40 p[3] p[4] + 48 p[3] p[5] + 25 p[4] + 60 p[4] p[5] + 36 p[5] - p[0] / - 4 p[1] - 9 p[2] - 16 p[3] - 25 p[4] - 36 p[5]) / / 3\1/2 (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] + 6 p[5] - 1) | / and in Maple notation ((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+10*p[0]*p[4]+12*p[0]*p[5]+4*p[1]^2 +12*p[1]*p[2]+16*p[1]*p[3]+20*p[1]*p[4]+24*p[1]*p[5]+9*p[2]^2+24*p[2]*p[3]+30*p [2]*p[4]+36*p[2]*p[5]+16*p[3]^2+40*p[3]*p[4]+48*p[3]*p[5]+25*p[4]^2+60*p[4]*p[5 ]+36*p[5]^2-p[0]-4*p[1]-9*p[2]-16*p[3]-25*p[4]-36*p[5])/(p[0]+2*p[1]+3*p[2]+4*p [3]+5*p[4]+6*p[5]-1)^3)^(1/2) The explicit expression for the, 3, -th scaled moment is 4 3 3 3 3 - (p[0] + 8 p[0] p[1] + 12 p[0] p[2] + 16 p[0] p[3] + 20 p[0] p[4] 3 2 2 2 2 + 24 p[0] p[5] + 24 p[0] p[1] + 72 p[0] p[1] p[2] + 96 p[0] p[1] p[3] 2 2 2 2 + 120 p[0] p[1] p[4] + 144 p[0] p[1] p[5] + 54 p[0] p[2] 2 2 2 + 144 p[0] p[2] p[3] + 180 p[0] p[2] p[4] + 216 p[0] p[2] p[5] 2 2 2 2 + 96 p[0] p[3] + 240 p[0] p[3] p[4] + 288 p[0] p[3] p[5] 2 2 2 2 2 3 + 150 p[0] p[4] + 360 p[0] p[4] p[5] + 216 p[0] p[5] + 32 p[0] p[1] 2 2 2 + 144 p[0] p[1] p[2] + 192 p[0] p[1] p[3] + 240 p[0] p[1] p[4] 2 2 + 288 p[0] p[1] p[5] + 216 p[0] p[1] p[2] + 576 p[0] p[1] p[2] p[3] 2 + 720 p[0] p[1] p[2] p[4] + 864 p[0] p[1] p[2] p[5] + 384 p[0] p[1] p[3] 2 + 960 p[0] p[1] p[3] p[4] + 1152 p[0] p[1] p[3] p[5] + 600 p[0] p[1] p[4] 2 3 + 1440 p[0] p[1] p[4] p[5] + 864 p[0] p[1] p[5] + 108 p[0] p[2] 2 2 2 + 432 p[0] p[2] p[3] + 540 p[0] p[2] p[4] + 648 p[0] p[2] p[5] 2 + 576 p[0] p[2] p[3] + 1440 p[0] p[2] p[3] p[4] 2 + 1728 p[0] p[2] p[3] p[5] + 900 p[0] p[2] p[4] 2 3 + 2160 p[0] p[2] p[4] p[5] + 1296 p[0] p[2] p[5] + 256 p[0] p[3] 2 2 2 + 960 p[0] p[3] p[4] + 1152 p[0] p[3] p[5] + 1200 p[0] p[3] p[4] 2 3 + 2880 p[0] p[3] p[4] p[5] + 1728 p[0] p[3] p[5] + 500 p[0] p[4] 2 2 3 4 + 1800 p[0] p[4] p[5] + 2160 p[0] p[4] p[5] + 864 p[0] p[5] + 16 p[1] 3 3 3 3 + 96 p[1] p[2] + 128 p[1] p[3] + 160 p[1] p[4] + 192 p[1] p[5] 2 2 2 2 + 216 p[1] p[2] + 576 p[1] p[2] p[3] + 720 p[1] p[2] p[4] 2 2 2 2 + 864 p[1] p[2] p[5] + 384 p[1] p[3] + 960 p[1] p[3] p[4] 2 2 2 2 + 1152 p[1] p[3] p[5] + 600 p[1] p[4] + 1440 p[1] p[4] p[5] 2 2 3 2 + 864 p[1] p[5] + 216 p[1] p[2] + 864 p[1] p[2] p[3] 2 2 2 + 1080 p[1] p[2] p[4] + 1296 p[1] p[2] p[5] + 1152 p[1] p[2] p[3] + 2880 p[1] p[2] p[3] p[4] + 3456 p[1] p[2] p[3] p[5] 2 2 + 1800 p[1] p[2] p[4] + 4320 p[1] p[2] p[4] p[5] + 2592 p[1] p[2] p[5] 3 2 2 + 512 p[1] p[3] + 1920 p[1] p[3] p[4] + 2304 p[1] p[3] p[5] 2 2 + 2400 p[1] p[3] p[4] + 5760 p[1] p[3] p[4] p[5] + 3456 p[1] p[3] p[5] 3 2 2 + 1000 p[1] p[4] + 3600 p[1] p[4] p[5] + 4320 p[1] p[4] p[5] 3 4 3 3 + 1728 p[1] p[5] + 81 p[2] + 432 p[2] p[3] + 540 p[2] p[4] 3 2 2 2 + 648 p[2] p[5] + 864 p[2] p[3] + 2160 p[2] p[3] p[4] 2 2 2 2 + 2592 p[2] p[3] p[5] + 1350 p[2] p[4] + 3240 p[2] p[4] p[5] 2 2 3 2 + 1944 p[2] p[5] + 768 p[2] p[3] + 2880 p[2] p[3] p[4] 2 2 + 3456 p[2] p[3] p[5] + 3600 p[2] p[3] p[4] + 8640 p[2] p[3] p[4] p[5] 2 3 2 + 5184 p[2] p[3] p[5] + 1500 p[2] p[4] + 5400 p[2] p[4] p[5] 2 3 4 3 + 6480 p[2] p[4] p[5] + 2592 p[2] p[5] + 256 p[3] + 1280 p[3] p[4] 3 2 2 2 + 1536 p[3] p[5] + 2400 p[3] p[4] + 5760 p[3] p[4] p[5] 2 2 3 2 + 3456 p[3] p[5] + 2000 p[3] p[4] + 7200 p[3] p[4] p[5] 2 3 4 3 + 8640 p[3] p[4] p[5] + 3456 p[3] p[5] + 625 p[4] + 3000 p[4] p[5] 2 2 3 4 3 + 5400 p[4] p[5] + 4320 p[4] p[5] + 1296 p[5] - 2 p[0] 2 2 2 2 - 16 p[0] p[1] - 30 p[0] p[2] - 48 p[0] p[3] - 70 p[0] p[4] 2 2 - 96 p[0] p[5] - 40 p[0] p[1] - 144 p[0] p[1] p[2] - 224 p[0] p[1] p[3] 2 - 320 p[0] p[1] p[4] - 432 p[0] p[1] p[5] - 126 p[0] p[2] - 384 p[0] p[2] p[3] - 540 p[0] p[2] p[4] - 720 p[0] p[2] p[5] 2 - 288 p[0] p[3] - 800 p[0] p[3] p[4] - 1056 p[0] p[3] p[5] 2 2 3 - 550 p[0] p[4] - 1440 p[0] p[4] p[5] - 936 p[0] p[5] - 32 p[1] 2 2 2 2 - 168 p[1] p[2] - 256 p[1] p[3] - 360 p[1] p[4] - 480 p[1] p[5] 2 - 288 p[1] p[2] - 864 p[1] p[2] p[3] - 1200 p[1] p[2] p[4] 2 - 1584 p[1] p[2] p[5] - 640 p[1] p[3] - 1760 p[1] p[3] p[4] 2 - 2304 p[1] p[3] p[5] - 1200 p[1] p[4] - 3120 p[1] p[4] p[5] 2 3 2 2 - 2016 p[1] p[5] - 162 p[2] - 720 p[2] p[3] - 990 p[2] p[4] 2 2 - 1296 p[2] p[5] - 1056 p[2] p[3] - 2880 p[2] p[3] p[4] 2 - 3744 p[2] p[3] p[5] - 1950 p[2] p[4] - 5040 p[2] p[4] p[5] 2 3 2 2 - 3240 p[2] p[5] - 512 p[3] - 2080 p[3] p[4] - 2688 p[3] p[5] 2 2 3 - 2800 p[3] p[4] - 7200 p[3] p[4] p[5] - 4608 p[3] p[5] - 1250 p[4] 2 2 3 2 - 4800 p[4] p[5] - 6120 p[4] p[5] - 2592 p[5] + 2 p[0] + 4 p[0] p[1] 2 - 6 p[0] p[2] - 32 p[0] p[3] - 80 p[0] p[4] - 156 p[0] p[5] + 12 p[1] 2 + 48 p[1] p[2] + 64 p[1] p[3] + 60 p[1] p[4] + 24 p[1] p[5] + 72 p[2] 2 + 264 p[2] p[3] + 390 p[2] p[4] + 504 p[2] p[5] + 272 p[3] 2 2 + 880 p[3] p[4] + 1248 p[3] p[5] + 750 p[4] + 2220 p[4] p[5] + 1692 p[5] / / - 2 p[0] + 4 p[1] + 24 p[2] + 64 p[3] + 130 p[4] + 228 p[5] + 1) / | / \ 5 / 2 (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] + 6 p[5] - 1) |(p[0] \ + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] + 10 p[0] p[4] + 12 p[0] p[5] 2 + 4 p[1] + 12 p[1] p[2] + 16 p[1] p[3] + 20 p[1] p[4] + 24 p[1] p[5] 2 2 + 9 p[2] + 24 p[2] p[3] + 30 p[2] p[4] + 36 p[2] p[5] + 16 p[3] 2 2 + 40 p[3] p[4] + 48 p[3] p[5] + 25 p[4] + 60 p[4] p[5] + 36 p[5] - p[0] / - 4 p[1] - 9 p[2] - 16 p[3] - 25 p[4] - 36 p[5]) / / 3\3/2\ (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] + 6 p[5] - 1) | | / / and in Maple notation -(p[0]^4+8*p[0]^3*p[1]+12*p[0]^3*p[2]+16*p[0]^3*p[3]+20*p[0]^3*p[4]+24*p[0]^3*p [5]+24*p[0]^2*p[1]^2+72*p[0]^2*p[1]*p[2]+96*p[0]^2*p[1]*p[3]+120*p[0]^2*p[1]*p[ 4]+144*p[0]^2*p[1]*p[5]+54*p[0]^2*p[2]^2+144*p[0]^2*p[2]*p[3]+180*p[0]^2*p[2]*p [4]+216*p[0]^2*p[2]*p[5]+96*p[0]^2*p[3]^2+240*p[0]^2*p[3]*p[4]+288*p[0]^2*p[3]* p[5]+150*p[0]^2*p[4]^2+360*p[0]^2*p[4]*p[5]+216*p[0]^2*p[5]^2+32*p[0]*p[1]^3+ 144*p[0]*p[1]^2*p[2]+192*p[0]*p[1]^2*p[3]+240*p[0]*p[1]^2*p[4]+288*p[0]*p[1]^2* p[5]+216*p[0]*p[1]*p[2]^2+576*p[0]*p[1]*p[2]*p[3]+720*p[0]*p[1]*p[2]*p[4]+864*p [0]*p[1]*p[2]*p[5]+384*p[0]*p[1]*p[3]^2+960*p[0]*p[1]*p[3]*p[4]+1152*p[0]*p[1]* p[3]*p[5]+600*p[0]*p[1]*p[4]^2+1440*p[0]*p[1]*p[4]*p[5]+864*p[0]*p[1]*p[5]^2+ 108*p[0]*p[2]^3+432*p[0]*p[2]^2*p[3]+540*p[0]*p[2]^2*p[4]+648*p[0]*p[2]^2*p[5]+ 576*p[0]*p[2]*p[3]^2+1440*p[0]*p[2]*p[3]*p[4]+1728*p[0]*p[2]*p[3]*p[5]+900*p[0] *p[2]*p[4]^2+2160*p[0]*p[2]*p[4]*p[5]+1296*p[0]*p[2]*p[5]^2+256*p[0]*p[3]^3+960 *p[0]*p[3]^2*p[4]+1152*p[0]*p[3]^2*p[5]+1200*p[0]*p[3]*p[4]^2+2880*p[0]*p[3]*p[ 4]*p[5]+1728*p[0]*p[3]*p[5]^2+500*p[0]*p[4]^3+1800*p[0]*p[4]^2*p[5]+2160*p[0]*p [4]*p[5]^2+864*p[0]*p[5]^3+16*p[1]^4+96*p[1]^3*p[2]+128*p[1]^3*p[3]+160*p[1]^3* p[4]+192*p[1]^3*p[5]+216*p[1]^2*p[2]^2+576*p[1]^2*p[2]*p[3]+720*p[1]^2*p[2]*p[4 ]+864*p[1]^2*p[2]*p[5]+384*p[1]^2*p[3]^2+960*p[1]^2*p[3]*p[4]+1152*p[1]^2*p[3]* p[5]+600*p[1]^2*p[4]^2+1440*p[1]^2*p[4]*p[5]+864*p[1]^2*p[5]^2+216*p[1]*p[2]^3+ 864*p[1]*p[2]^2*p[3]+1080*p[1]*p[2]^2*p[4]+1296*p[1]*p[2]^2*p[5]+1152*p[1]*p[2] *p[3]^2+2880*p[1]*p[2]*p[3]*p[4]+3456*p[1]*p[2]*p[3]*p[5]+1800*p[1]*p[2]*p[4]^2 +4320*p[1]*p[2]*p[4]*p[5]+2592*p[1]*p[2]*p[5]^2+512*p[1]*p[3]^3+1920*p[1]*p[3]^ 2*p[4]+2304*p[1]*p[3]^2*p[5]+2400*p[1]*p[3]*p[4]^2+5760*p[1]*p[3]*p[4]*p[5]+ 3456*p[1]*p[3]*p[5]^2+1000*p[1]*p[4]^3+3600*p[1]*p[4]^2*p[5]+4320*p[1]*p[4]*p[5 ]^2+1728*p[1]*p[5]^3+81*p[2]^4+432*p[2]^3*p[3]+540*p[2]^3*p[4]+648*p[2]^3*p[5]+ 864*p[2]^2*p[3]^2+2160*p[2]^2*p[3]*p[4]+2592*p[2]^2*p[3]*p[5]+1350*p[2]^2*p[4]^ 2+3240*p[2]^2*p[4]*p[5]+1944*p[2]^2*p[5]^2+768*p[2]*p[3]^3+2880*p[2]*p[3]^2*p[4 ]+3456*p[2]*p[3]^2*p[5]+3600*p[2]*p[3]*p[4]^2+8640*p[2]*p[3]*p[4]*p[5]+5184*p[2 ]*p[3]*p[5]^2+1500*p[2]*p[4]^3+5400*p[2]*p[4]^2*p[5]+6480*p[2]*p[4]*p[5]^2+2592 *p[2]*p[5]^3+256*p[3]^4+1280*p[3]^3*p[4]+1536*p[3]^3*p[5]+2400*p[3]^2*p[4]^2+ 5760*p[3]^2*p[4]*p[5]+3456*p[3]^2*p[5]^2+2000*p[3]*p[4]^3+7200*p[3]*p[4]^2*p[5] +8640*p[3]*p[4]*p[5]^2+3456*p[3]*p[5]^3+625*p[4]^4+3000*p[4]^3*p[5]+5400*p[4]^2 *p[5]^2+4320*p[4]*p[5]^3+1296*p[5]^4-2*p[0]^3-16*p[0]^2*p[1]-30*p[0]^2*p[2]-48* p[0]^2*p[3]-70*p[0]^2*p[4]-96*p[0]^2*p[5]-40*p[0]*p[1]^2-144*p[0]*p[1]*p[2]-224 *p[0]*p[1]*p[3]-320*p[0]*p[1]*p[4]-432*p[0]*p[1]*p[5]-126*p[0]*p[2]^2-384*p[0]* p[2]*p[3]-540*p[0]*p[2]*p[4]-720*p[0]*p[2]*p[5]-288*p[0]*p[3]^2-800*p[0]*p[3]*p [4]-1056*p[0]*p[3]*p[5]-550*p[0]*p[4]^2-1440*p[0]*p[4]*p[5]-936*p[0]*p[5]^2-32* p[1]^3-168*p[1]^2*p[2]-256*p[1]^2*p[3]-360*p[1]^2*p[4]-480*p[1]^2*p[5]-288*p[1] *p[2]^2-864*p[1]*p[2]*p[3]-1200*p[1]*p[2]*p[4]-1584*p[1]*p[2]*p[5]-640*p[1]*p[3 ]^2-1760*p[1]*p[3]*p[4]-2304*p[1]*p[3]*p[5]-1200*p[1]*p[4]^2-3120*p[1]*p[4]*p[5 ]-2016*p[1]*p[5]^2-162*p[2]^3-720*p[2]^2*p[3]-990*p[2]^2*p[4]-1296*p[2]^2*p[5]-\ 1056*p[2]*p[3]^2-2880*p[2]*p[3]*p[4]-3744*p[2]*p[3]*p[5]-1950*p[2]*p[4]^2-5040* p[2]*p[4]*p[5]-3240*p[2]*p[5]^2-512*p[3]^3-2080*p[3]^2*p[4]-2688*p[3]^2*p[5]-\ 2800*p[3]*p[4]^2-7200*p[3]*p[4]*p[5]-4608*p[3]*p[5]^2-1250*p[4]^3-4800*p[4]^2*p [5]-6120*p[4]*p[5]^2-2592*p[5]^3+2*p[0]^2+4*p[0]*p[1]-6*p[0]*p[2]-32*p[0]*p[3]-\ 80*p[0]*p[4]-156*p[0]*p[5]+12*p[1]^2+48*p[1]*p[2]+64*p[1]*p[3]+60*p[1]*p[4]+24* p[1]*p[5]+72*p[2]^2+264*p[2]*p[3]+390*p[2]*p[4]+504*p[2]*p[5]+272*p[3]^2+880*p[ 3]*p[4]+1248*p[3]*p[5]+750*p[4]^2+2220*p[4]*p[5]+1692*p[5]^2-2*p[0]+4*p[1]+24*p [2]+64*p[3]+130*p[4]+228*p[5]+1)/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]+6*p[5]-1)^5/ ((p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+10*p[0]*p[4]+12*p[0]*p[5]+4*p[1]^2 +12*p[1]*p[2]+16*p[1]*p[3]+20*p[1]*p[4]+24*p[1]*p[5]+9*p[2]^2+24*p[2]*p[3]+30*p [2]*p[4]+36*p[2]*p[5]+16*p[3]^2+40*p[3]*p[4]+48*p[3]*p[5]+25*p[4]^2+60*p[4]*p[5 ]+36*p[5]^2-p[0]-4*p[1]-9*p[2]-16*p[3]-25*p[4]-36*p[5])/(p[0]+2*p[1]+3*p[2]+4*p [3]+5*p[4]+6*p[5]-1)^3)^(3/2) The explicit expression for the, 4, -th scaled moment is 6 5 5 5 5 (p[0] + 12 p[0] p[1] + 18 p[0] p[2] + 24 p[0] p[3] + 30 p[0] p[4] 5 4 2 4 + 36 p[0] p[5] + 60 p[0] p[1] + 180 p[0] p[1] p[2] 4 4 4 + 240 p[0] p[1] p[3] + 300 p[0] p[1] p[4] + 360 p[0] p[1] p[5] 4 2 4 4 + 135 p[0] p[2] + 360 p[0] p[2] p[3] + 450 p[0] p[2] p[4] 4 4 2 4 + 540 p[0] p[2] p[5] + 240 p[0] p[3] + 600 p[0] p[3] p[4] 4 4 2 4 + 720 p[0] p[3] p[5] + 375 p[0] p[4] + 900 p[0] p[4] p[5] 4 2 3 3 3 2 + 540 p[0] p[5] + 160 p[0] p[1] + 720 p[0] p[1] p[2] 3 2 3 2 3 2 + 960 p[0] p[1] p[3] + 1200 p[0] p[1] p[4] + 1440 p[0] p[1] p[5] 3 2 3 + 1080 p[0] p[1] p[2] + 2880 p[0] p[1] p[2] p[3] 3 3 + 3600 p[0] p[1] p[2] p[4] + 4320 p[0] p[1] p[2] p[5] 3 2 3 + 1920 p[0] p[1] p[3] + 4800 p[0] p[1] p[3] p[4] 3 3 2 + 5760 p[0] p[1] p[3] p[5] + 3000 p[0] p[1] p[4] 3 3 2 3 3 + 7200 p[0] p[1] p[4] p[5] + 4320 p[0] p[1] p[5] + 540 p[0] p[2] 3 2 3 2 3 2 + 2160 p[0] p[2] p[3] + 2700 p[0] p[2] p[4] + 3240 p[0] p[2] p[5] 3 2 3 + 2880 p[0] p[2] p[3] + 7200 p[0] p[2] p[3] p[4] 3 3 2 + 8640 p[0] p[2] p[3] p[5] + 4500 p[0] p[2] p[4] 3 3 2 3 3 + 10800 p[0] p[2] p[4] p[5] + 6480 p[0] p[2] p[5] + 1280 p[0] p[3] 3 2 3 2 3 2 + 4800 p[0] p[3] p[4] + 5760 p[0] p[3] p[5] + 6000 p[0] p[3] p[4] 3 3 2 3 3 + 14400 p[0] p[3] p[4] p[5] + 8640 p[0] p[3] p[5] + 2500 p[0] p[4] 3 2 3 2 3 3 + 9000 p[0] p[4] p[5] + 10800 p[0] p[4] p[5] + 4320 p[0] p[5] 2 4 2 3 2 3 + 240 p[0] p[1] + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] 2 3 2 3 2 2 2 + 2400 p[0] p[1] p[4] + 2880 p[0] p[1] p[5] + 3240 p[0] p[1] p[2] 2 2 2 2 + 8640 p[0] p[1] p[2] p[3] + 10800 p[0] p[1] p[2] p[4] 2 2 2 2 2 + 12960 p[0] p[1] p[2] p[5] + 5760 p[0] p[1] p[3] 2 2 2 2 + 14400 p[0] p[1] p[3] p[4] + 17280 p[0] p[1] p[3] p[5] 2 2 2 2 2 + 9000 p[0] p[1] p[4] + 21600 p[0] p[1] p[4] p[5] 2 2 2 2 3 + 12960 p[0] p[1] p[5] + 3240 p[0] p[1] p[2] 2 2 2 2 + 12960 p[0] p[1] p[2] p[3] + 16200 p[0] p[1] p[2] p[4] 2 2 2 2 + 19440 p[0] p[1] p[2] p[5] + 17280 p[0] p[1] p[2] p[3] 2 2 + 43200 p[0] p[1] p[2] p[3] p[4] + 51840 p[0] p[1] p[2] p[3] p[5] 2 2 2 + 27000 p[0] p[1] p[2] p[4] + 64800 p[0] p[1] p[2] p[4] p[5] 2 2 2 3 + 38880 p[0] p[1] p[2] p[5] + 7680 p[0] p[1] p[3] 2 2 2 2 + 28800 p[0] p[1] p[3] p[4] + 34560 p[0] p[1] p[3] p[5] 2 2 2 + 36000 p[0] p[1] p[3] p[4] + 86400 p[0] p[1] p[3] p[4] p[5] 2 2 2 3 + 51840 p[0] p[1] p[3] p[5] + 15000 p[0] p[1] p[4] 2 2 2 2 + 54000 p[0] p[1] p[4] p[5] + 64800 p[0] p[1] p[4] p[5] 2 3 2 4 2 3 + 25920 p[0] p[1] p[5] + 1215 p[0] p[2] + 6480 p[0] p[2] p[3] 2 3 2 3 2 2 2 + 8100 p[0] p[2] p[4] + 9720 p[0] p[2] p[5] + 12960 p[0] p[2] p[3] 2 2 2 2 + 32400 p[0] p[2] p[3] p[4] + 38880 p[0] p[2] p[3] p[5] 2 2 2 2 2 + 20250 p[0] p[2] p[4] + 48600 p[0] p[2] p[4] p[5] 2 2 2 2 3 + 29160 p[0] p[2] p[5] + 11520 p[0] p[2] p[3] 2 2 2 2 + 43200 p[0] p[2] p[3] p[4] + 51840 p[0] p[2] p[3] p[5] 2 2 2 + 54000 p[0] p[2] p[3] p[4] + 129600 p[0] p[2] p[3] p[4] p[5] 2 2 2 3 + 77760 p[0] p[2] p[3] p[5] + 22500 p[0] p[2] p[4] 2 2 2 2 + 81000 p[0] p[2] p[4] p[5] + 97200 p[0] p[2] p[4] p[5] 2 3 2 4 2 3 + 38880 p[0] p[2] p[5] + 3840 p[0] p[3] + 19200 p[0] p[3] p[4] 2 3 2 2 2 + 23040 p[0] p[3] p[5] + 36000 p[0] p[3] p[4] 2 2 2 2 2 + 86400 p[0] p[3] p[4] p[5] + 51840 p[0] p[3] p[5] 2 3 2 2 + 30000 p[0] p[3] p[4] + 108000 p[0] p[3] p[4] p[5] 2 2 2 3 2 4 + 129600 p[0] p[3] p[4] p[5] + 51840 p[0] p[3] p[5] + 9375 p[0] p[4] 2 3 2 2 2 + 45000 p[0] p[4] p[5] + 81000 p[0] p[4] p[5] 2 3 2 4 5 + 64800 p[0] p[4] p[5] + 19440 p[0] p[5] + 192 p[0] p[1] 4 4 4 + 1440 p[0] p[1] p[2] + 1920 p[0] p[1] p[3] + 2400 p[0] p[1] p[4] 4 3 2 + 2880 p[0] p[1] p[5] + 4320 p[0] p[1] p[2] 3 3 + 11520 p[0] p[1] p[2] p[3] + 14400 p[0] p[1] p[2] p[4] 3 3 2 + 17280 p[0] p[1] p[2] p[5] + 7680 p[0] p[1] p[3] 3 3 + 19200 p[0] p[1] p[3] p[4] + 23040 p[0] p[1] p[3] p[5] 3 2 3 + 12000 p[0] p[1] p[4] + 28800 p[0] p[1] p[4] p[5] 3 2 2 3 + 17280 p[0] p[1] p[5] + 6480 p[0] p[1] p[2] 2 2 2 2 + 25920 p[0] p[1] p[2] p[3] + 32400 p[0] p[1] p[2] p[4] 2 2 2 2 + 38880 p[0] p[1] p[2] p[5] + 34560 p[0] p[1] p[2] p[3] 2 2 + 86400 p[0] p[1] p[2] p[3] p[4] + 103680 p[0] p[1] p[2] p[3] p[5] 2 2 2 + 54000 p[0] p[1] p[2] p[4] + 129600 p[0] p[1] p[2] p[4] p[5] 2 2 2 3 + 77760 p[0] p[1] p[2] p[5] + 15360 p[0] p[1] p[3] 2 2 2 2 + 57600 p[0] p[1] p[3] p[4] + 69120 p[0] p[1] p[3] p[5] 2 2 2 + 72000 p[0] p[1] p[3] p[4] + 172800 p[0] p[1] p[3] p[4] p[5] 2 2 2 3 + 103680 p[0] p[1] p[3] p[5] + 30000 p[0] p[1] p[4] 2 2 2 2 + 108000 p[0] p[1] p[4] p[5] + 129600 p[0] p[1] p[4] p[5] 2 3 4 + 51840 p[0] p[1] p[5] + 4860 p[0] p[1] p[2] 3 3 + 25920 p[0] p[1] p[2] p[3] + 32400 p[0] p[1] p[2] p[4] 3 2 2 + 38880 p[0] p[1] p[2] p[5] + 51840 p[0] p[1] p[2] p[3] 2 2 + 129600 p[0] p[1] p[2] p[3] p[4] + 155520 p[0] p[1] p[2] p[3] p[5] 2 2 2 + 81000 p[0] p[1] p[2] p[4] + 194400 p[0] p[1] p[2] p[4] p[5] 2 2 3 + 116640 p[0] p[1] p[2] p[5] + 46080 p[0] p[1] p[2] p[3] 2 2 + 172800 p[0] p[1] p[2] p[3] p[4] + 207360 p[0] p[1] p[2] p[3] p[5] 2 + 216000 p[0] p[1] p[2] p[3] p[4] + 518400 p[0] p[1] p[2] p[3] p[4] p[5] 2 3 + 311040 p[0] p[1] p[2] p[3] p[5] + 90000 p[0] p[1] p[2] p[4] 2 2 + 324000 p[0] p[1] p[2] p[4] p[5] + 388800 p[0] p[1] p[2] p[4] p[5] 3 4 + 155520 p[0] p[1] p[2] p[5] + 15360 p[0] p[1] p[3] 3 3 + 76800 p[0] p[1] p[3] p[4] + 92160 p[0] p[1] p[3] p[5] 2 2 2 + 144000 p[0] p[1] p[3] p[4] + 345600 p[0] p[1] p[3] p[4] p[5] 2 2 3 + 207360 p[0] p[1] p[3] p[5] + 120000 p[0] p[1] p[3] p[4] 2 2 + 432000 p[0] p[1] p[3] p[4] p[5] + 518400 p[0] p[1] p[3] p[4] p[5] 3 4 + 207360 p[0] p[1] p[3] p[5] + 37500 p[0] p[1] p[4] 3 2 2 + 180000 p[0] p[1] p[4] p[5] + 324000 p[0] p[1] p[4] p[5] 3 4 5 + 259200 p[0] p[1] p[4] p[5] + 77760 p[0] p[1] p[5] + 1458 p[0] p[2] 4 4 4 + 9720 p[0] p[2] p[3] + 12150 p[0] p[2] p[4] + 14580 p[0] p[2] p[5] 3 2 3 + 25920 p[0] p[2] p[3] + 64800 p[0] p[2] p[3] p[4] 3 3 2 + 77760 p[0] p[2] p[3] p[5] + 40500 p[0] p[2] p[4] 3 3 2 + 97200 p[0] p[2] p[4] p[5] + 58320 p[0] p[2] p[5] 2 3 2 2 + 34560 p[0] p[2] p[3] + 129600 p[0] p[2] p[3] p[4] 2 2 2 2 + 155520 p[0] p[2] p[3] p[5] + 162000 p[0] p[2] p[3] p[4] 2 2 2 + 388800 p[0] p[2] p[3] p[4] p[5] + 233280 p[0] p[2] p[3] p[5] 2 3 2 2 + 67500 p[0] p[2] p[4] + 243000 p[0] p[2] p[4] p[5] 2 2 2 3 + 291600 p[0] p[2] p[4] p[5] + 116640 p[0] p[2] p[5] 4 3 + 23040 p[0] p[2] p[3] + 115200 p[0] p[2] p[3] p[4] 3 2 2 + 138240 p[0] p[2] p[3] p[5] + 216000 p[0] p[2] p[3] p[4] 2 2 2 + 518400 p[0] p[2] p[3] p[4] p[5] + 311040 p[0] p[2] p[3] p[5] 3 2 + 180000 p[0] p[2] p[3] p[4] + 648000 p[0] p[2] p[3] p[4] p[5] 2 3 + 777600 p[0] p[2] p[3] p[4] p[5] + 311040 p[0] p[2] p[3] p[5] 4 3 + 56250 p[0] p[2] p[4] + 270000 p[0] p[2] p[4] p[5] 2 2 3 + 486000 p[0] p[2] p[4] p[5] + 388800 p[0] p[2] p[4] p[5] 4 5 4 + 116640 p[0] p[2] p[5] + 6144 p[0] p[3] + 38400 p[0] p[3] p[4] 4 3 2 + 46080 p[0] p[3] p[5] + 96000 p[0] p[3] p[4] 3 3 2 + 230400 p[0] p[3] p[4] p[5] + 138240 p[0] p[3] p[5] 2 3 2 2 + 120000 p[0] p[3] p[4] + 432000 p[0] p[3] p[4] p[5] 2 2 2 3 + 518400 p[0] p[3] p[4] p[5] + 207360 p[0] p[3] p[5] 4 3 + 75000 p[0] p[3] p[4] + 360000 p[0] p[3] p[4] p[5] 2 2 3 + 648000 p[0] p[3] p[4] p[5] + 518400 p[0] p[3] p[4] p[5] 4 5 4 + 155520 p[0] p[3] p[5] + 18750 p[0] p[4] + 112500 p[0] p[4] p[5] 3 2 2 3 + 270000 p[0] p[4] p[5] + 324000 p[0] p[4] p[5] 4 5 6 5 + 194400 p[0] p[4] p[5] + 46656 p[0] p[5] + 64 p[1] + 576 p[1] p[2] 5 5 5 4 2 + 768 p[1] p[3] + 960 p[1] p[4] + 1152 p[1] p[5] + 2160 p[1] p[2] 4 4 4 + 5760 p[1] p[2] p[3] + 7200 p[1] p[2] p[4] + 8640 p[1] p[2] p[5] 4 2 4 4 + 3840 p[1] p[3] + 9600 p[1] p[3] p[4] + 11520 p[1] p[3] p[5] 4 2 4 4 2 + 6000 p[1] p[4] + 14400 p[1] p[4] p[5] + 8640 p[1] p[5] 3 3 3 2 3 2 + 4320 p[1] p[2] + 17280 p[1] p[2] p[3] + 21600 p[1] p[2] p[4] 3 2 3 2 + 25920 p[1] p[2] p[5] + 23040 p[1] p[2] p[3] 3 3 + 57600 p[1] p[2] p[3] p[4] + 69120 p[1] p[2] p[3] p[5] 3 2 3 + 36000 p[1] p[2] p[4] + 86400 p[1] p[2] p[4] p[5] 3 2 3 3 3 2 + 51840 p[1] p[2] p[5] + 10240 p[1] p[3] + 38400 p[1] p[3] p[4] 3 2 3 2 + 46080 p[1] p[3] p[5] + 48000 p[1] p[3] p[4] 3 3 2 3 3 + 115200 p[1] p[3] p[4] p[5] + 69120 p[1] p[3] p[5] + 20000 p[1] p[4] 3 2 3 2 3 3 + 72000 p[1] p[4] p[5] + 86400 p[1] p[4] p[5] + 34560 p[1] p[5] 2 4 2 3 2 3 + 4860 p[1] p[2] + 25920 p[1] p[2] p[3] + 32400 p[1] p[2] p[4] 2 3 2 2 2 + 38880 p[1] p[2] p[5] + 51840 p[1] p[2] p[3] 2 2 2 2 + 129600 p[1] p[2] p[3] p[4] + 155520 p[1] p[2] p[3] p[5] 2 2 2 2 2 + 81000 p[1] p[2] p[4] + 194400 p[1] p[2] p[4] p[5] 2 2 2 2 3 + 116640 p[1] p[2] p[5] + 46080 p[1] p[2] p[3] 2 2 2 2 + 172800 p[1] p[2] p[3] p[4] + 207360 p[1] p[2] p[3] p[5] 2 2 2 + 216000 p[1] p[2] p[3] p[4] + 518400 p[1] p[2] p[3] p[4] p[5] 2 2 2 3 + 311040 p[1] p[2] p[3] p[5] + 90000 p[1] p[2] p[4] 2 2 2 2 + 324000 p[1] p[2] p[4] p[5] + 388800 p[1] p[2] p[4] p[5] 2 3 2 4 2 3 + 155520 p[1] p[2] p[5] + 15360 p[1] p[3] + 76800 p[1] p[3] p[4] 2 3 2 2 2 + 92160 p[1] p[3] p[5] + 144000 p[1] p[3] p[4] 2 2 2 2 2 + 345600 p[1] p[3] p[4] p[5] + 207360 p[1] p[3] p[5] 2 3 2 2 + 120000 p[1] p[3] p[4] + 432000 p[1] p[3] p[4] p[5] 2 2 2 3 + 518400 p[1] p[3] p[4] p[5] + 207360 p[1] p[3] p[5] 2 4 2 3 2 2 2 + 37500 p[1] p[4] + 180000 p[1] p[4] p[5] + 324000 p[1] p[4] p[5] 2 3 2 4 5 + 259200 p[1] p[4] p[5] + 77760 p[1] p[5] + 2916 p[1] p[2] 4 4 4 + 19440 p[1] p[2] p[3] + 24300 p[1] p[2] p[4] + 29160 p[1] p[2] p[5] 3 2 3 + 51840 p[1] p[2] p[3] + 129600 p[1] p[2] p[3] p[4] 3 3 2 + 155520 p[1] p[2] p[3] p[5] + 81000 p[1] p[2] p[4] 3 3 2 + 194400 p[1] p[2] p[4] p[5] + 116640 p[1] p[2] p[5] 2 3 2 2 + 69120 p[1] p[2] p[3] + 259200 p[1] p[2] p[3] p[4] 2 2 2 2 + 311040 p[1] p[2] p[3] p[5] + 324000 p[1] p[2] p[3] p[4] 2 2 2 + 777600 p[1] p[2] p[3] p[4] p[5] + 466560 p[1] p[2] p[3] p[5] 2 3 2 2 + 135000 p[1] p[2] p[4] + 486000 p[1] p[2] p[4] p[5] 2 2 2 3 + 583200 p[1] p[2] p[4] p[5] + 233280 p[1] p[2] p[5] 4 3 + 46080 p[1] p[2] p[3] + 230400 p[1] p[2] p[3] p[4] 3 2 2 + 276480 p[1] p[2] p[3] p[5] + 432000 p[1] p[2] p[3] p[4] 2 2 2 + 1036800 p[1] p[2] p[3] p[4] p[5] + 622080 p[1] p[2] p[3] p[5] 3 2 + 360000 p[1] p[2] p[3] p[4] + 1296000 p[1] p[2] p[3] p[4] p[5] 2 3 + 1555200 p[1] p[2] p[3] p[4] p[5] + 622080 p[1] p[2] p[3] p[5] 4 3 + 112500 p[1] p[2] p[4] + 540000 p[1] p[2] p[4] p[5] 2 2 3 + 972000 p[1] p[2] p[4] p[5] + 777600 p[1] p[2] p[4] p[5] 4 5 4 + 233280 p[1] p[2] p[5] + 12288 p[1] p[3] + 76800 p[1] p[3] p[4] 4 3 2 + 92160 p[1] p[3] p[5] + 192000 p[1] p[3] p[4] 3 3 2 + 460800 p[1] p[3] p[4] p[5] + 276480 p[1] p[3] p[5] 2 3 2 2 + 240000 p[1] p[3] p[4] + 864000 p[1] p[3] p[4] p[5] 2 2 2 3 + 1036800 p[1] p[3] p[4] p[5] + 414720 p[1] p[3] p[5] 4 3 + 150000 p[1] p[3] p[4] + 720000 p[1] p[3] p[4] p[5] 2 2 3 + 1296000 p[1] p[3] p[4] p[5] + 1036800 p[1] p[3] p[4] p[5] 4 5 4 + 311040 p[1] p[3] p[5] + 37500 p[1] p[4] + 225000 p[1] p[4] p[5] 3 2 2 3 + 540000 p[1] p[4] p[5] + 648000 p[1] p[4] p[5] 4 5 6 5 + 388800 p[1] p[4] p[5] + 93312 p[1] p[5] + 729 p[2] + 5832 p[2] p[3] 5 5 4 2 + 7290 p[2] p[4] + 8748 p[2] p[5] + 19440 p[2] p[3] 4 4 4 2 + 48600 p[2] p[3] p[4] + 58320 p[2] p[3] p[5] + 30375 p[2] p[4] 4 4 2 3 3 + 72900 p[2] p[4] p[5] + 43740 p[2] p[5] + 34560 p[2] p[3] 3 2 3 2 + 129600 p[2] p[3] p[4] + 155520 p[2] p[3] p[5] 3 2 3 + 162000 p[2] p[3] p[4] + 388800 p[2] p[3] p[4] p[5] 3 2 3 3 3 2 + 233280 p[2] p[3] p[5] + 67500 p[2] p[4] + 243000 p[2] p[4] p[5] 3 2 3 3 2 4 + 291600 p[2] p[4] p[5] + 116640 p[2] p[5] + 34560 p[2] p[3] 2 3 2 3 + 172800 p[2] p[3] p[4] + 207360 p[2] p[3] p[5] 2 2 2 2 2 + 324000 p[2] p[3] p[4] + 777600 p[2] p[3] p[4] p[5] 2 2 2 2 3 + 466560 p[2] p[3] p[5] + 270000 p[2] p[3] p[4] 2 2 2 2 + 972000 p[2] p[3] p[4] p[5] + 1166400 p[2] p[3] p[4] p[5] 2 3 2 4 2 3 + 466560 p[2] p[3] p[5] + 84375 p[2] p[4] + 405000 p[2] p[4] p[5] 2 2 2 2 3 2 4 + 729000 p[2] p[4] p[5] + 583200 p[2] p[4] p[5] + 174960 p[2] p[5] 5 4 4 + 18432 p[2] p[3] + 115200 p[2] p[3] p[4] + 138240 p[2] p[3] p[5] 3 2 3 + 288000 p[2] p[3] p[4] + 691200 p[2] p[3] p[4] p[5] 3 2 2 3 + 414720 p[2] p[3] p[5] + 360000 p[2] p[3] p[4] 2 2 2 2 + 1296000 p[2] p[3] p[4] p[5] + 1555200 p[2] p[3] p[4] p[5] 2 3 4 + 622080 p[2] p[3] p[5] + 225000 p[2] p[3] p[4] 3 2 2 + 1080000 p[2] p[3] p[4] p[5] + 1944000 p[2] p[3] p[4] p[5] 3 4 5 + 1555200 p[2] p[3] p[4] p[5] + 466560 p[2] p[3] p[5] + 56250 p[2] p[4] 4 3 2 + 337500 p[2] p[4] p[5] + 810000 p[2] p[4] p[5] 2 3 4 5 + 972000 p[2] p[4] p[5] + 583200 p[2] p[4] p[5] + 139968 p[2] p[5] 6 5 5 4 2 + 4096 p[3] + 30720 p[3] p[4] + 36864 p[3] p[5] + 96000 p[3] p[4] 4 4 2 3 3 + 230400 p[3] p[4] p[5] + 138240 p[3] p[5] + 160000 p[3] p[4] 3 2 3 2 3 3 + 576000 p[3] p[4] p[5] + 691200 p[3] p[4] p[5] + 276480 p[3] p[5] 2 4 2 3 2 2 2 + 150000 p[3] p[4] + 720000 p[3] p[4] p[5] + 1296000 p[3] p[4] p[5] 2 3 2 4 5 + 1036800 p[3] p[4] p[5] + 311040 p[3] p[5] + 75000 p[3] p[4] 4 3 2 + 450000 p[3] p[4] p[5] + 1080000 p[3] p[4] p[5] 2 3 4 5 + 1296000 p[3] p[4] p[5] + 777600 p[3] p[4] p[5] + 186624 p[3] p[5] 6 5 4 2 + 15625 p[4] + 112500 p[4] p[5] + 337500 p[4] p[5] 3 3 2 4 5 + 540000 p[4] p[5] + 486000 p[4] p[5] + 233280 p[4] p[5] 6 4 4 4 + 46656 p[5] - 6 p[0] p[1] - 18 p[0] p[2] - 36 p[0] p[3] 4 4 3 2 3 - 60 p[0] p[4] - 90 p[0] p[5] - 48 p[0] p[1] - 216 p[0] p[1] p[2] 3 3 3 - 384 p[0] p[1] p[3] - 600 p[0] p[1] p[4] - 864 p[0] p[1] p[5] 3 2 3 3 - 216 p[0] p[2] - 720 p[0] p[2] p[3] - 1080 p[0] p[2] p[4] 3 3 2 3 - 1512 p[0] p[2] p[5] - 576 p[0] p[3] - 1680 p[0] p[3] p[4] 3 3 2 3 - 2304 p[0] p[3] p[5] - 1200 p[0] p[4] - 3240 p[0] p[4] p[5] 3 2 2 3 2 2 - 2160 p[0] p[5] - 144 p[0] p[1] - 864 p[0] p[1] p[2] 2 2 2 2 2 2 - 1440 p[0] p[1] p[3] - 2160 p[0] p[1] p[4] - 3024 p[0] p[1] p[5] 2 2 2 - 1620 p[0] p[1] p[2] - 5184 p[0] p[1] p[2] p[3] 2 2 - 7560 p[0] p[1] p[2] p[4] - 10368 p[0] p[1] p[2] p[5] 2 2 2 - 4032 p[0] p[1] p[3] - 11520 p[0] p[1] p[3] p[4] 2 2 2 - 15552 p[0] p[1] p[3] p[5] - 8100 p[0] p[1] p[4] 2 2 2 2 3 - 21600 p[0] p[1] p[4] p[5] - 14256 p[0] p[1] p[5] - 972 p[0] p[2] 2 2 2 2 2 2 - 4536 p[0] p[2] p[3] - 6480 p[0] p[2] p[4] - 8748 p[0] p[2] p[5] 2 2 2 - 6912 p[0] p[2] p[3] - 19440 p[0] p[2] p[3] p[4] 2 2 2 - 25920 p[0] p[2] p[3] p[5] - 13500 p[0] p[2] p[4] 2 2 2 2 3 - 35640 p[0] p[2] p[4] p[5] - 23328 p[0] p[2] p[5] - 3456 p[0] p[3] 2 2 2 2 2 2 - 14400 p[0] p[3] p[4] - 19008 p[0] p[3] p[5] - 19800 p[0] p[3] p[4] 2 2 2 2 3 - 51840 p[0] p[3] p[4] p[5] - 33696 p[0] p[3] p[5] - 9000 p[0] p[4] 2 2 2 2 2 3 - 35100 p[0] p[4] p[5] - 45360 p[0] p[4] p[5] - 19440 p[0] p[5] 4 3 3 - 192 p[0] p[1] - 1440 p[0] p[1] p[2] - 2304 p[0] p[1] p[3] 3 3 2 2 - 3360 p[0] p[1] p[4] - 4608 p[0] p[1] p[5] - 3888 p[0] p[1] p[2] 2 2 - 12096 p[0] p[1] p[2] p[3] - 17280 p[0] p[1] p[2] p[4] 2 2 2 - 23328 p[0] p[1] p[2] p[5] - 9216 p[0] p[1] p[3] 2 2 - 25920 p[0] p[1] p[3] p[4] - 34560 p[0] p[1] p[3] p[5] 2 2 2 - 18000 p[0] p[1] p[4] - 47520 p[0] p[1] p[4] p[5] 2 2 3 - 31104 p[0] p[1] p[5] - 4536 p[0] p[1] p[2] 2 2 - 20736 p[0] p[1] p[2] p[3] - 29160 p[0] p[1] p[2] p[4] 2 2 - 38880 p[0] p[1] p[2] p[5] - 31104 p[0] p[1] p[2] p[3] - 86400 p[0] p[1] p[2] p[3] p[4] - 114048 p[0] p[1] p[2] p[3] p[5] 2 - 59400 p[0] p[1] p[2] p[4] - 155520 p[0] p[1] p[2] p[4] p[5] 2 3 - 101088 p[0] p[1] p[2] p[5] - 15360 p[0] p[1] p[3] 2 2 - 63360 p[0] p[1] p[3] p[4] - 82944 p[0] p[1] p[3] p[5] 2 - 86400 p[0] p[1] p[3] p[4] - 224640 p[0] p[1] p[3] p[4] p[5] 2 3 - 145152 p[0] p[1] p[3] p[5] - 39000 p[0] p[1] p[4] 2 2 - 151200 p[0] p[1] p[4] p[5] - 194400 p[0] p[1] p[4] p[5] 3 4 3 - 82944 p[0] p[1] p[5] - 1944 p[0] p[2] - 11664 p[0] p[2] p[3] 3 3 2 2 - 16200 p[0] p[2] p[4] - 21384 p[0] p[2] p[5] - 25920 p[0] p[2] p[3] 2 2 - 71280 p[0] p[2] p[3] p[4] - 93312 p[0] p[2] p[3] p[5] 2 2 2 - 48600 p[0] p[2] p[4] - 126360 p[0] p[2] p[4] p[5] 2 2 3 - 81648 p[0] p[2] p[5] - 25344 p[0] p[2] p[3] 2 2 - 103680 p[0] p[2] p[3] p[4] - 134784 p[0] p[2] p[3] p[5] 2 - 140400 p[0] p[2] p[3] p[4] - 362880 p[0] p[2] p[3] p[4] p[5] 2 3 - 233280 p[0] p[2] p[3] p[5] - 63000 p[0] p[2] p[4] 2 2 - 243000 p[0] p[2] p[4] p[5] - 311040 p[0] p[2] p[4] p[5] 3 4 3 - 132192 p[0] p[2] p[5] - 9216 p[0] p[3] - 49920 p[0] p[3] p[4] 3 2 2 - 64512 p[0] p[3] p[5] - 100800 p[0] p[3] p[4] 2 2 2 - 259200 p[0] p[3] p[4] p[5] - 165888 p[0] p[3] p[5] 3 2 - 90000 p[0] p[3] p[4] - 345600 p[0] p[3] p[4] p[5] 2 3 4 - 440640 p[0] p[3] p[4] p[5] - 186624 p[0] p[3] p[5] - 30000 p[0] p[4] 3 2 2 - 153000 p[0] p[4] p[5] - 291600 p[0] p[4] p[5] 3 4 5 4 - 246240 p[0] p[4] p[5] - 77760 p[0] p[5] - 96 p[1] - 864 p[1] p[2] 4 4 4 3 2 - 1344 p[1] p[3] - 1920 p[1] p[4] - 2592 p[1] p[5] - 3024 p[1] p[2] 3 3 3 - 9216 p[1] p[2] p[3] - 12960 p[1] p[2] p[4] - 17280 p[1] p[2] p[5] 3 2 3 3 - 6912 p[1] p[3] - 19200 p[1] p[3] p[4] - 25344 p[1] p[3] p[5] 3 2 3 3 2 - 13200 p[1] p[4] - 34560 p[1] p[4] p[5] - 22464 p[1] p[5] 2 3 2 2 2 2 - 5184 p[1] p[2] - 23328 p[1] p[2] p[3] - 32400 p[1] p[2] p[4] 2 2 2 2 - 42768 p[1] p[2] p[5] - 34560 p[1] p[2] p[3] 2 2 - 95040 p[1] p[2] p[3] p[4] - 124416 p[1] p[2] p[3] p[5] 2 2 2 - 64800 p[1] p[2] p[4] - 168480 p[1] p[2] p[4] p[5] 2 2 2 3 2 2 - 108864 p[1] p[2] p[5] - 16896 p[1] p[3] - 69120 p[1] p[3] p[4] 2 2 2 2 - 89856 p[1] p[3] p[5] - 93600 p[1] p[3] p[4] 2 2 2 - 241920 p[1] p[3] p[4] p[5] - 155520 p[1] p[3] p[5] 2 3 2 2 2 2 - 42000 p[1] p[4] - 162000 p[1] p[4] p[5] - 207360 p[1] p[4] p[5] 2 3 4 3 - 88128 p[1] p[5] - 4374 p[1] p[2] - 25920 p[1] p[2] p[3] 3 3 2 2 - 35640 p[1] p[2] p[4] - 46656 p[1] p[2] p[5] - 57024 p[1] p[2] p[3] 2 2 - 155520 p[1] p[2] p[3] p[4] - 202176 p[1] p[2] p[3] p[5] 2 2 2 - 105300 p[1] p[2] p[4] - 272160 p[1] p[2] p[4] p[5] 2 2 3 - 174960 p[1] p[2] p[5] - 55296 p[1] p[2] p[3] 2 2 - 224640 p[1] p[2] p[3] p[4] - 290304 p[1] p[2] p[3] p[5] 2 - 302400 p[1] p[2] p[3] p[4] - 777600 p[1] p[2] p[3] p[4] p[5] 2 3 - 497664 p[1] p[2] p[3] p[5] - 135000 p[1] p[2] p[4] 2 2 - 518400 p[1] p[2] p[4] p[5] - 660960 p[1] p[2] p[4] p[5] 3 4 3 - 279936 p[1] p[2] p[5] - 19968 p[1] p[3] - 107520 p[1] p[3] p[4] 3 2 2 - 138240 p[1] p[3] p[5] - 216000 p[1] p[3] p[4] 2 2 2 - 552960 p[1] p[3] p[4] p[5] - 352512 p[1] p[3] p[5] 3 2 - 192000 p[1] p[3] p[4] - 734400 p[1] p[3] p[4] p[5] 2 3 4 - 933120 p[1] p[3] p[4] p[5] - 393984 p[1] p[3] p[5] - 63750 p[1] p[4] 3 2 2 - 324000 p[1] p[4] p[5] - 615600 p[1] p[4] p[5] 3 4 5 - 518400 p[1] p[4] p[5] - 163296 p[1] p[5] - 1458 p[2] 4 4 4 - 10692 p[2] p[3] - 14580 p[2] p[4] - 18954 p[2] p[5] 3 2 3 3 - 31104 p[2] p[3] - 84240 p[2] p[3] p[4] - 108864 p[2] p[3] p[5] 3 2 3 3 2 - 56700 p[2] p[4] - 145800 p[2] p[4] p[5] - 93312 p[2] p[5] 2 3 2 2 2 2 - 44928 p[2] p[3] - 181440 p[2] p[3] p[4] - 233280 p[2] p[3] p[5] 2 2 2 - 243000 p[2] p[3] p[4] - 622080 p[2] p[3] p[4] p[5] 2 2 2 3 2 2 - 396576 p[2] p[3] p[5] - 108000 p[2] p[4] - 413100 p[2] p[4] p[5] 2 2 2 3 4 - 524880 p[2] p[4] p[5] - 221616 p[2] p[5] - 32256 p[2] p[3] 3 3 - 172800 p[2] p[3] p[4] - 221184 p[2] p[3] p[5] 2 2 2 - 345600 p[2] p[3] p[4] - 881280 p[2] p[3] p[4] p[5] 2 2 3 - 559872 p[2] p[3] p[5] - 306000 p[2] p[3] p[4] 2 2 - 1166400 p[2] p[3] p[4] p[5] - 1477440 p[2] p[3] p[4] p[5] 3 4 3 - 622080 p[2] p[3] p[5] - 101250 p[2] p[4] - 513000 p[2] p[4] p[5] 2 2 3 4 - 972000 p[2] p[4] p[5] - 816480 p[2] p[4] p[5] - 256608 p[2] p[5] 5 4 4 3 2 - 9216 p[3] - 61440 p[3] p[4] - 78336 p[3] p[5] - 163200 p[3] p[4] 3 3 2 2 3 - 414720 p[3] p[4] p[5] - 262656 p[3] p[5] - 216000 p[3] p[4] 2 2 2 2 2 3 - 820800 p[3] p[4] p[5] - 1036800 p[3] p[4] p[5] - 435456 p[3] p[5] 4 3 2 2 - 142500 p[3] p[4] - 720000 p[3] p[4] p[5] - 1360800 p[3] p[4] p[5] 3 4 5 - 1140480 p[3] p[4] p[5] - 357696 p[3] p[5] - 37500 p[4] 4 3 2 2 3 - 236250 p[4] p[5] - 594000 p[4] p[5] - 745200 p[4] p[5] 4 5 4 3 - 466560 p[4] p[5] - 116640 p[5] - p[0] - 36 p[0] p[1] 3 3 3 3 - 114 p[0] p[2] - 256 p[0] p[3] - 480 p[0] p[4] - 804 p[0] p[5] 2 2 2 2 - 156 p[0] p[1] - 720 p[0] p[1] p[2] - 1440 p[0] p[1] p[3] 2 2 2 2 - 2580 p[0] p[1] p[4] - 4248 p[0] p[1] p[5] - 648 p[0] p[2] 2 2 2 - 2232 p[0] p[2] p[3] - 3690 p[0] p[2] p[4] - 5832 p[0] p[2] p[5] 2 2 2 2 - 1680 p[0] p[3] - 5040 p[0] p[3] p[4] - 7488 p[0] p[3] p[5] 2 2 2 2 2 - 3450 p[0] p[4] - 9540 p[0] p[4] p[5] - 6156 p[0] p[5] 3 2 2 - 224 p[0] p[1] - 1296 p[0] p[1] p[2] - 2400 p[0] p[1] p[3] 2 2 2 - 4200 p[0] p[1] p[4] - 6912 p[0] p[1] p[5] - 2052 p[0] p[1] p[2] - 6624 p[0] p[1] p[2] p[3] - 10800 p[0] p[1] p[2] p[4] 2 - 17280 p[0] p[1] p[2] p[5] - 4608 p[0] p[1] p[3] - 13440 p[0] p[1] p[3] p[4] - 20160 p[0] p[1] p[3] p[5] 2 2 - 8700 p[0] p[1] p[4] - 23760 p[0] p[1] p[4] p[5] - 14688 p[0] p[1] p[5] 3 2 2 - 918 p[0] p[2] - 3888 p[0] p[2] p[3] - 5940 p[0] p[2] p[4] 2 2 - 9396 p[0] p[2] p[5] - 4608 p[0] p[2] p[3] - 12240 p[0] p[2] p[3] p[4] 2 - 18144 p[0] p[2] p[3] p[5] - 6750 p[0] p[2] p[4] 2 3 - 17280 p[0] p[2] p[4] p[5] - 9072 p[0] p[2] p[5] - 1408 p[0] p[3] 2 2 2 - 4320 p[0] p[3] p[4] - 5760 p[0] p[3] p[5] - 2400 p[0] p[3] p[4] 2 3 - 2880 p[0] p[3] p[4] p[5] + 2592 p[0] p[3] p[5] + 1000 p[0] p[4] 2 2 3 + 8100 p[0] p[4] p[5] + 18360 p[0] p[4] p[5] + 12096 p[0] p[5] 4 3 3 3 - 96 p[1] - 624 p[1] p[2] - 1088 p[1] p[3] - 1920 p[1] p[4] 3 2 2 2 - 3264 p[1] p[5] - 1188 p[1] p[2] - 3456 p[1] p[2] p[3] 2 2 2 2 - 5760 p[1] p[2] p[4] - 9936 p[1] p[2] p[5] - 1920 p[1] p[3] 2 2 2 2 - 5280 p[1] p[3] p[4] - 8640 p[1] p[3] p[5] - 2700 p[1] p[4] 2 2 2 3 - 7200 p[1] p[4] p[5] - 3456 p[1] p[5] - 648 p[1] p[2] 2 2 2 - 1728 p[1] p[2] p[3] - 2700 p[1] p[2] p[4] - 5832 p[1] p[2] p[5] 2 + 576 p[1] p[2] p[3] + 4320 p[1] p[2] p[3] p[4] 2 + 3456 p[1] p[2] p[3] p[5] + 7200 p[1] p[2] p[4] 2 3 + 21600 p[1] p[2] p[4] p[5] + 19440 p[1] p[2] p[5] + 2304 p[1] p[3] 2 2 2 + 13440 p[1] p[3] p[4] + 18432 p[1] p[3] p[5] + 26400 p[1] p[3] p[4] 2 3 + 77760 p[1] p[3] p[4] p[5] + 60480 p[1] p[3] p[5] + 16500 p[1] p[4] 2 2 3 + 73800 p[1] p[4] p[5] + 112320 p[1] p[4] p[5] + 57024 p[1] p[5] 4 3 3 3 + 81 p[2] + 1512 p[2] p[3] + 2430 p[2] p[4] + 2592 p[2] p[5] 2 2 2 2 + 6480 p[2] p[3] + 21600 p[2] p[3] p[4] + 28512 p[2] p[3] p[5] 2 2 2 2 2 + 18900 p[2] p[4] + 53460 p[2] p[4] p[5] + 39852 p[2] p[5] 3 2 2 + 9984 p[2] p[3] + 48960 p[2] p[3] p[4] + 67392 p[2] p[3] p[5] 2 + 81000 p[2] p[3] p[4] + 228960 p[2] p[3] p[4] p[5] 2 3 2 + 165888 p[2] p[3] p[5] + 44250 p[2] p[4] + 189000 p[2] p[4] p[5] 2 3 4 + 272160 p[2] p[4] p[5] + 130896 p[2] p[5] + 5120 p[3] 3 3 2 2 + 32640 p[3] p[4] + 45312 p[3] p[5] + 78000 p[3] p[4] 2 2 2 3 + 218880 p[3] p[4] p[5] + 155520 p[3] p[5] + 82000 p[3] p[4] 2 2 3 + 345600 p[3] p[4] p[5] + 488160 p[3] p[4] p[5] + 229824 p[3] p[5] 4 3 2 2 3 + 31875 p[4] + 178500 p[4] p[5] + 375300 p[4] p[5] + 349920 p[4] p[5] 4 3 2 2 2 + 121824 p[5] - 7 p[0] - 6 p[0] p[1] + 27 p[0] p[2] + 36 p[0] p[3] 2 2 2 - 45 p[0] p[4] - 306 p[0] p[5] + 180 p[0] p[1] + 1164 p[0] p[1] p[2] + 2352 p[0] p[1] p[3] + 3780 p[0] p[1] p[4] + 5208 p[0] p[1] p[5] 2 + 1791 p[0] p[2] + 7416 p[0] p[2] p[3] + 12630 p[0] p[2] p[4] 2 + 19044 p[0] p[2] p[5] + 7824 p[0] p[3] + 27240 p[0] p[3] p[4] 2 + 42192 p[0] p[3] p[5] + 24075 p[0] p[4] + 75660 p[0] p[4] p[5] 2 3 2 2 + 60084 p[0] p[5] + 208 p[1] + 1500 p[1] p[2] + 2880 p[1] p[3] 2 2 2 + 4740 p[1] p[4] + 6960 p[1] p[5] + 3186 p[1] p[2] + 11856 p[1] p[2] p[3] + 19740 p[1] p[2] p[4] + 30024 p[1] p[2] p[5] 2 + 10944 p[1] p[3] + 36720 p[1] p[3] p[4] + 56832 p[1] p[3] p[5] 2 2 3 + 31050 p[1] p[4] + 97080 p[1] p[4] p[5] + 76464 p[1] p[5] + 1701 p[2] 2 2 2 2 + 8244 p[2] p[3] + 13095 p[2] p[4] + 19926 p[2] p[5] + 11952 p[2] p[3] 2 + 36120 p[2] p[3] p[4] + 54576 p[2] p[3] p[5] + 26475 p[2] p[4] 2 3 2 + 79380 p[2] p[4] p[5] + 59292 p[2] p[5] + 4352 p[3] + 16080 p[3] p[4] 2 2 + 22464 p[3] p[5] + 15300 p[3] p[4] + 36240 p[3] p[4] p[5] 2 3 2 2 + 17856 p[3] p[5] + 625 p[4] - 10050 p[4] p[5] - 39420 p[4] p[5] 3 2 - 34992 p[5] + 12 p[0] + 84 p[0] p[1] + 270 p[0] p[2] + 720 p[0] p[3] 2 + 1620 p[0] p[4] + 3204 p[0] p[5] - 36 p[1] - 408 p[1] p[2] 2 - 768 p[1] p[3] - 900 p[1] p[4] - 456 p[1] p[5] - 1062 p[2] 2 - 4824 p[2] p[3] - 8310 p[2] p[4] - 12240 p[2] p[5] - 5808 p[3] 2 - 21120 p[3] p[4] - 33120 p[3] p[5] - 19800 p[4] - 63780 p[4] p[5] 2 - 52308 p[5] - 5 p[0] - 48 p[1] - 183 p[2] - 488 p[3] - 1065 p[4] / 2 - 2040 p[5]) / ((p[0] + 4 p[0] p[1] + 6 p[0] p[2] + 8 p[0] p[3] / 2 + 10 p[0] p[4] + 12 p[0] p[5] + 4 p[1] + 12 p[1] p[2] + 16 p[1] p[3] 2 + 20 p[1] p[4] + 24 p[1] p[5] + 9 p[2] + 24 p[2] p[3] + 30 p[2] p[4] 2 2 + 36 p[2] p[5] + 16 p[3] + 40 p[3] p[4] + 48 p[3] p[5] + 25 p[4] 2 + 60 p[4] p[5] + 36 p[5] - p[0] - 4 p[1] - 9 p[2] - 16 p[3] - 25 p[4] 2 - 36 p[5]) (p[0] + 2 p[1] + 3 p[2] + 4 p[3] + 5 p[4] + 6 p[5] - 1)) and in Maple notation (p[0]^6+12*p[0]^5*p[1]+18*p[0]^5*p[2]+24*p[0]^5*p[3]+30*p[0]^5*p[4]+36*p[0]^5*p [5]+60*p[0]^4*p[1]^2+180*p[0]^4*p[1]*p[2]+240*p[0]^4*p[1]*p[3]+300*p[0]^4*p[1]* p[4]+360*p[0]^4*p[1]*p[5]+135*p[0]^4*p[2]^2+360*p[0]^4*p[2]*p[3]+450*p[0]^4*p[2 ]*p[4]+540*p[0]^4*p[2]*p[5]+240*p[0]^4*p[3]^2+600*p[0]^4*p[3]*p[4]+720*p[0]^4*p [3]*p[5]+375*p[0]^4*p[4]^2+900*p[0]^4*p[4]*p[5]+540*p[0]^4*p[5]^2+160*p[0]^3*p[ 1]^3+720*p[0]^3*p[1]^2*p[2]+960*p[0]^3*p[1]^2*p[3]+1200*p[0]^3*p[1]^2*p[4]+1440 *p[0]^3*p[1]^2*p[5]+1080*p[0]^3*p[1]*p[2]^2+2880*p[0]^3*p[1]*p[2]*p[3]+3600*p[0 ]^3*p[1]*p[2]*p[4]+4320*p[0]^3*p[1]*p[2]*p[5]+1920*p[0]^3*p[1]*p[3]^2+4800*p[0] ^3*p[1]*p[3]*p[4]+5760*p[0]^3*p[1]*p[3]*p[5]+3000*p[0]^3*p[1]*p[4]^2+7200*p[0]^ 3*p[1]*p[4]*p[5]+4320*p[0]^3*p[1]*p[5]^2+540*p[0]^3*p[2]^3+2160*p[0]^3*p[2]^2*p [3]+2700*p[0]^3*p[2]^2*p[4]+3240*p[0]^3*p[2]^2*p[5]+2880*p[0]^3*p[2]*p[3]^2+ 7200*p[0]^3*p[2]*p[3]*p[4]+8640*p[0]^3*p[2]*p[3]*p[5]+4500*p[0]^3*p[2]*p[4]^2+ 10800*p[0]^3*p[2]*p[4]*p[5]+6480*p[0]^3*p[2]*p[5]^2+1280*p[0]^3*p[3]^3+4800*p[0 ]^3*p[3]^2*p[4]+5760*p[0]^3*p[3]^2*p[5]+6000*p[0]^3*p[3]*p[4]^2+14400*p[0]^3*p[ 3]*p[4]*p[5]+8640*p[0]^3*p[3]*p[5]^2+2500*p[0]^3*p[4]^3+9000*p[0]^3*p[4]^2*p[5] +10800*p[0]^3*p[4]*p[5]^2+4320*p[0]^3*p[5]^3+240*p[0]^2*p[1]^4+1440*p[0]^2*p[1] ^3*p[2]+1920*p[0]^2*p[1]^3*p[3]+2400*p[0]^2*p[1]^3*p[4]+2880*p[0]^2*p[1]^3*p[5] +3240*p[0]^2*p[1]^2*p[2]^2+8640*p[0]^2*p[1]^2*p[2]*p[3]+10800*p[0]^2*p[1]^2*p[2 ]*p[4]+12960*p[0]^2*p[1]^2*p[2]*p[5]+5760*p[0]^2*p[1]^2*p[3]^2+14400*p[0]^2*p[1 ]^2*p[3]*p[4]+17280*p[0]^2*p[1]^2*p[3]*p[5]+9000*p[0]^2*p[1]^2*p[4]^2+21600*p[0 ]^2*p[1]^2*p[4]*p[5]+12960*p[0]^2*p[1]^2*p[5]^2+3240*p[0]^2*p[1]*p[2]^3+12960*p [0]^2*p[1]*p[2]^2*p[3]+16200*p[0]^2*p[1]*p[2]^2*p[4]+19440*p[0]^2*p[1]*p[2]^2*p [5]+17280*p[0]^2*p[1]*p[2]*p[3]^2+43200*p[0]^2*p[1]*p[2]*p[3]*p[4]+51840*p[0]^2 *p[1]*p[2]*p[3]*p[5]+27000*p[0]^2*p[1]*p[2]*p[4]^2+64800*p[0]^2*p[1]*p[2]*p[4]* p[5]+38880*p[0]^2*p[1]*p[2]*p[5]^2+7680*p[0]^2*p[1]*p[3]^3+28800*p[0]^2*p[1]*p[ 3]^2*p[4]+34560*p[0]^2*p[1]*p[3]^2*p[5]+36000*p[0]^2*p[1]*p[3]*p[4]^2+86400*p[0 ]^2*p[1]*p[3]*p[4]*p[5]+51840*p[0]^2*p[1]*p[3]*p[5]^2+15000*p[0]^2*p[1]*p[4]^3+ 54000*p[0]^2*p[1]*p[4]^2*p[5]+64800*p[0]^2*p[1]*p[4]*p[5]^2+25920*p[0]^2*p[1]*p [5]^3+1215*p[0]^2*p[2]^4+6480*p[0]^2*p[2]^3*p[3]+8100*p[0]^2*p[2]^3*p[4]+9720*p [0]^2*p[2]^3*p[5]+12960*p[0]^2*p[2]^2*p[3]^2+32400*p[0]^2*p[2]^2*p[3]*p[4]+ 38880*p[0]^2*p[2]^2*p[3]*p[5]+20250*p[0]^2*p[2]^2*p[4]^2+48600*p[0]^2*p[2]^2*p[ 4]*p[5]+29160*p[0]^2*p[2]^2*p[5]^2+11520*p[0]^2*p[2]*p[3]^3+43200*p[0]^2*p[2]*p [3]^2*p[4]+51840*p[0]^2*p[2]*p[3]^2*p[5]+54000*p[0]^2*p[2]*p[3]*p[4]^2+129600*p [0]^2*p[2]*p[3]*p[4]*p[5]+77760*p[0]^2*p[2]*p[3]*p[5]^2+22500*p[0]^2*p[2]*p[4]^ 3+81000*p[0]^2*p[2]*p[4]^2*p[5]+97200*p[0]^2*p[2]*p[4]*p[5]^2+38880*p[0]^2*p[2] *p[5]^3+3840*p[0]^2*p[3]^4+19200*p[0]^2*p[3]^3*p[4]+23040*p[0]^2*p[3]^3*p[5]+ 36000*p[0]^2*p[3]^2*p[4]^2+86400*p[0]^2*p[3]^2*p[4]*p[5]+51840*p[0]^2*p[3]^2*p[ 5]^2+30000*p[0]^2*p[3]*p[4]^3+108000*p[0]^2*p[3]*p[4]^2*p[5]+129600*p[0]^2*p[3] *p[4]*p[5]^2+51840*p[0]^2*p[3]*p[5]^3+9375*p[0]^2*p[4]^4+45000*p[0]^2*p[4]^3*p[ 5]+81000*p[0]^2*p[4]^2*p[5]^2+64800*p[0]^2*p[4]*p[5]^3+19440*p[0]^2*p[5]^4+192* p[0]*p[1]^5+1440*p[0]*p[1]^4*p[2]+1920*p[0]*p[1]^4*p[3]+2400*p[0]*p[1]^4*p[4]+ 2880*p[0]*p[1]^4*p[5]+4320*p[0]*p[1]^3*p[2]^2+11520*p[0]*p[1]^3*p[2]*p[3]+14400 *p[0]*p[1]^3*p[2]*p[4]+17280*p[0]*p[1]^3*p[2]*p[5]+7680*p[0]*p[1]^3*p[3]^2+ 19200*p[0]*p[1]^3*p[3]*p[4]+23040*p[0]*p[1]^3*p[3]*p[5]+12000*p[0]*p[1]^3*p[4]^ 2+28800*p[0]*p[1]^3*p[4]*p[5]+17280*p[0]*p[1]^3*p[5]^2+6480*p[0]*p[1]^2*p[2]^3+ 25920*p[0]*p[1]^2*p[2]^2*p[3]+32400*p[0]*p[1]^2*p[2]^2*p[4]+38880*p[0]*p[1]^2*p [2]^2*p[5]+34560*p[0]*p[1]^2*p[2]*p[3]^2+86400*p[0]*p[1]^2*p[2]*p[3]*p[4]+ 103680*p[0]*p[1]^2*p[2]*p[3]*p[5]+54000*p[0]*p[1]^2*p[2]*p[4]^2+129600*p[0]*p[1 ]^2*p[2]*p[4]*p[5]+77760*p[0]*p[1]^2*p[2]*p[5]^2+15360*p[0]*p[1]^2*p[3]^3+57600 *p[0]*p[1]^2*p[3]^2*p[4]+69120*p[0]*p[1]^2*p[3]^2*p[5]+72000*p[0]*p[1]^2*p[3]*p [4]^2+172800*p[0]*p[1]^2*p[3]*p[4]*p[5]+103680*p[0]*p[1]^2*p[3]*p[5]^2+30000*p[ 0]*p[1]^2*p[4]^3+108000*p[0]*p[1]^2*p[4]^2*p[5]+129600*p[0]*p[1]^2*p[4]*p[5]^2+ 51840*p[0]*p[1]^2*p[5]^3+4860*p[0]*p[1]*p[2]^4+25920*p[0]*p[1]*p[2]^3*p[3]+ 32400*p[0]*p[1]*p[2]^3*p[4]+38880*p[0]*p[1]*p[2]^3*p[5]+51840*p[0]*p[1]*p[2]^2* p[3]^2+129600*p[0]*p[1]*p[2]^2*p[3]*p[4]+155520*p[0]*p[1]*p[2]^2*p[3]*p[5]+ 81000*p[0]*p[1]*p[2]^2*p[4]^2+194400*p[0]*p[1]*p[2]^2*p[4]*p[5]+116640*p[0]*p[1 ]*p[2]^2*p[5]^2+46080*p[0]*p[1]*p[2]*p[3]^3+172800*p[0]*p[1]*p[2]*p[3]^2*p[4]+ 207360*p[0]*p[1]*p[2]*p[3]^2*p[5]+216000*p[0]*p[1]*p[2]*p[3]*p[4]^2+518400*p[0] *p[1]*p[2]*p[3]*p[4]*p[5]+311040*p[0]*p[1]*p[2]*p[3]*p[5]^2+90000*p[0]*p[1]*p[2 ]*p[4]^3+324000*p[0]*p[1]*p[2]*p[4]^2*p[5]+388800*p[0]*p[1]*p[2]*p[4]*p[5]^2+ 155520*p[0]*p[1]*p[2]*p[5]^3+15360*p[0]*p[1]*p[3]^4+76800*p[0]*p[1]*p[3]^3*p[4] +92160*p[0]*p[1]*p[3]^3*p[5]+144000*p[0]*p[1]*p[3]^2*p[4]^2+345600*p[0]*p[1]*p[ 3]^2*p[4]*p[5]+207360*p[0]*p[1]*p[3]^2*p[5]^2+120000*p[0]*p[1]*p[3]*p[4]^3+ 432000*p[0]*p[1]*p[3]*p[4]^2*p[5]+518400*p[0]*p[1]*p[3]*p[4]*p[5]^2+207360*p[0] *p[1]*p[3]*p[5]^3+37500*p[0]*p[1]*p[4]^4+180000*p[0]*p[1]*p[4]^3*p[5]+324000*p[ 0]*p[1]*p[4]^2*p[5]^2+259200*p[0]*p[1]*p[4]*p[5]^3+77760*p[0]*p[1]*p[5]^4+1458* p[0]*p[2]^5+9720*p[0]*p[2]^4*p[3]+12150*p[0]*p[2]^4*p[4]+14580*p[0]*p[2]^4*p[5] +25920*p[0]*p[2]^3*p[3]^2+64800*p[0]*p[2]^3*p[3]*p[4]+77760*p[0]*p[2]^3*p[3]*p[ 5]+40500*p[0]*p[2]^3*p[4]^2+97200*p[0]*p[2]^3*p[4]*p[5]+58320*p[0]*p[2]^3*p[5]^ 2+34560*p[0]*p[2]^2*p[3]^3+129600*p[0]*p[2]^2*p[3]^2*p[4]+155520*p[0]*p[2]^2*p[ 3]^2*p[5]+162000*p[0]*p[2]^2*p[3]*p[4]^2+388800*p[0]*p[2]^2*p[3]*p[4]*p[5]+ 233280*p[0]*p[2]^2*p[3]*p[5]^2+67500*p[0]*p[2]^2*p[4]^3+243000*p[0]*p[2]^2*p[4] ^2*p[5]+291600*p[0]*p[2]^2*p[4]*p[5]^2+116640*p[0]*p[2]^2*p[5]^3+23040*p[0]*p[2 ]*p[3]^4+115200*p[0]*p[2]*p[3]^3*p[4]+138240*p[0]*p[2]*p[3]^3*p[5]+216000*p[0]* p[2]*p[3]^2*p[4]^2+518400*p[0]*p[2]*p[3]^2*p[4]*p[5]+311040*p[0]*p[2]*p[3]^2*p[ 5]^2+180000*p[0]*p[2]*p[3]*p[4]^3+648000*p[0]*p[2]*p[3]*p[4]^2*p[5]+777600*p[0] *p[2]*p[3]*p[4]*p[5]^2+311040*p[0]*p[2]*p[3]*p[5]^3+56250*p[0]*p[2]*p[4]^4+ 270000*p[0]*p[2]*p[4]^3*p[5]+486000*p[0]*p[2]*p[4]^2*p[5]^2+388800*p[0]*p[2]*p[ 4]*p[5]^3+116640*p[0]*p[2]*p[5]^4+6144*p[0]*p[3]^5+38400*p[0]*p[3]^4*p[4]+46080 *p[0]*p[3]^4*p[5]+96000*p[0]*p[3]^3*p[4]^2+230400*p[0]*p[3]^3*p[4]*p[5]+138240* p[0]*p[3]^3*p[5]^2+120000*p[0]*p[3]^2*p[4]^3+432000*p[0]*p[3]^2*p[4]^2*p[5]+ 518400*p[0]*p[3]^2*p[4]*p[5]^2+207360*p[0]*p[3]^2*p[5]^3+75000*p[0]*p[3]*p[4]^4 +360000*p[0]*p[3]*p[4]^3*p[5]+648000*p[0]*p[3]*p[4]^2*p[5]^2+518400*p[0]*p[3]*p [4]*p[5]^3+155520*p[0]*p[3]*p[5]^4+18750*p[0]*p[4]^5+112500*p[0]*p[4]^4*p[5]+ 270000*p[0]*p[4]^3*p[5]^2+324000*p[0]*p[4]^2*p[5]^3+194400*p[0]*p[4]*p[5]^4+ 46656*p[0]*p[5]^5+64*p[1]^6+576*p[1]^5*p[2]+768*p[1]^5*p[3]+960*p[1]^5*p[4]+ 1152*p[1]^5*p[5]+2160*p[1]^4*p[2]^2+5760*p[1]^4*p[2]*p[3]+7200*p[1]^4*p[2]*p[4] 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3186*p[1]*p[2]^2+11856*p[1]*p[2]*p[3]+19740*p[1]*p[2]*p[4]+30024*p[1]*p[2]*p[5] +10944*p[1]*p[3]^2+36720*p[1]*p[3]*p[4]+56832*p[1]*p[3]*p[5]+31050*p[1]*p[4]^2+ 97080*p[1]*p[4]*p[5]+76464*p[1]*p[5]^2+1701*p[2]^3+8244*p[2]^2*p[3]+13095*p[2]^ 2*p[4]+19926*p[2]^2*p[5]+11952*p[2]*p[3]^2+36120*p[2]*p[3]*p[4]+54576*p[2]*p[3] *p[5]+26475*p[2]*p[4]^2+79380*p[2]*p[4]*p[5]+59292*p[2]*p[5]^2+4352*p[3]^3+ 16080*p[3]^2*p[4]+22464*p[3]^2*p[5]+15300*p[3]*p[4]^2+36240*p[3]*p[4]*p[5]+ 17856*p[3]*p[5]^2+625*p[4]^3-10050*p[4]^2*p[5]-39420*p[4]*p[5]^2-34992*p[5]^3+ 12*p[0]^2+84*p[0]*p[1]+270*p[0]*p[2]+720*p[0]*p[3]+1620*p[0]*p[4]+3204*p[0]*p[5 ]-36*p[1]^2-408*p[1]*p[2]-768*p[1]*p[3]-900*p[1]*p[4]-456*p[1]*p[5]-1062*p[2]^2 -4824*p[2]*p[3]-8310*p[2]*p[4]-12240*p[2]*p[5]-5808*p[3]^2-21120*p[3]*p[4]-\ 33120*p[3]*p[5]-19800*p[4]^2-63780*p[4]*p[5]-52308*p[5]^2-5*p[0]-48*p[1]-183*p[ 2]-488*p[3]-1065*p[4]-2040*p[5])/(p[0]^2+4*p[0]*p[1]+6*p[0]*p[2]+8*p[0]*p[3]+10 *p[0]*p[4]+12*p[0]*p[5]+4*p[1]^2+12*p[1]*p[2]+16*p[1]*p[3]+20*p[1]*p[4]+24*p[1] *p[5]+9*p[2]^2+24*p[2]*p[3]+30*p[2]*p[4]+36*p[2]*p[5]+16*p[3]^2+40*p[3]*p[4]+48 *p[3]*p[5]+25*p[4]^2+60*p[4]*p[5]+36*p[5]^2-p[0]-4*p[1]-9*p[2]-16*p[3]-25*p[4]-\ 36*p[5])^2/(p[0]+2*p[1]+3*p[2]+4*p[3]+5*p[4]+6*p[5]-1) This took, 0.595, seconds.