The Expected Duration, Variance, and all the moments up to the, 10, of the Duration of a Markov Process on the Integers from n to 0 where at each location the particle moves from n to n-1 with probability a^n\ , and stays where it is with probability 1-a^n By Shalosh B. Ekhad Theorem : Consider a walker, on the discrete positive line, starting at n, a\ nd goes left, to n-1 with prob. a^n, and remains at n with prob. 1-a^n The expectation number of moves until she makes it to 0 is n (1/a) - 1 - ---------- a - 1 and in Maple notation -((1/a)^n-1)/(a-1) The variance of the number of moves until she makes it to 0 is n n ((1/a) - 1) ((1/a) - a) - ------------------------- (a - 1) (a + 1) and in Maple notation -((1/a)^n-1)*((1/a)^n-a)/(a-1)/(a+1) The skewness is / n 2 n 2 \1/2 | (2 a (1/a) - a + 2 (1/a) + a - 1) (a - 1) (a + 1)| |- -----------------------------------------------------| | n n 2 2 | \ ((1/a) - 1) ((1/a) - a) (a + a + 1) / and in Maple notation (-1/((1/a)^n-1)*(2*a*(1/a)^n-a^2+2*(1/a)^n+a-1)^2/((1/a)^n-a)*(a-1)*(a+1)/(a^2+ a+1)^2)^(1/2) as n goes to infinity, the limiting skewness is / 3\1/2 | 4 (a - 1) (a + 1) | |- ------------------| | 2 2 | \ (a + a + 1) / and in Maple notation (-4*(a-1)*(a+1)^3/(a^2+a+1)^2)^(1/2) Here is a plot from a=0 to a=1 2 *HHHHHHHHHHH + HHHHHHHHH + HHHHHHH + HHHHHH + HHHHHH 1.5 HHHHHH + HHHHH + HHHHH + HHHHH + HHHH 1 + HHHHH + HHHH + HHHH + HHH + HHH 0.5 HHH + HH + H + H + H +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+--+--+--* 0 0.2 0.4 0.6 0.8 1 The kurtosis is n 2 4 5 n 6 3 n 2 4 n 5 - (3 ((1/a) ) a - 3 a (1/a) + a + 3 a ((1/a) ) + 6 a (1/a) - 8 a n 2 2 3 n 4 n 2 2 n - 6 ((1/a) ) a + 15 a (1/a) + 4 a - 9 ((1/a) ) a + 3 a (1/a) 3 n 2 n 2 n / - 6 a - 9 ((1/a) ) + 6 a (1/a) - 10 a + 9 (1/a) + 2 a - 1) / ( / n n 2 2 ((1/a) - 1) ((1/a) - a) (a + a + 1) (a + 1)) and in Maple notation -1/((1/a)^n-1)/((1/a)^n-a)*(3*((1/a)^n)^2*a^4-3*a^5*(1/a)^n+a^6+3*a^3*((1/a)^n) ^2+6*a^4*(1/a)^n-8*a^5-6*((1/a)^n)^2*a^2+15*a^3*(1/a)^n+4*a^4-9*((1/a)^n)^2*a+3 *a^2*(1/a)^n-6*a^3-9*((1/a)^n)^2+6*a*(1/a)^n-10*a^2+9*(1/a)^n+2*a-1)/(a^2+a+1)/ (a^2+1) as n goes to infinity, the limiting kurtosis is 2 3 (a - 3) - ---------- 2 a + 1 and in Maple notation -3*(a^2-3)/(a^2+1) Here is a plot from a=0 to a=1 9 *HHHHHHHHH + HHHHHHH + HHHHHH 8 + HHHH + HHHH + HHHH + HHH 7 + HHH + HHH + HHH 6 + HHHH + HHHH + HHHH 5 + HHHH + HHHH + HHHH + HHH 4 + HHHH + HHHH + HHHHH 3 +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+--+--+*** 0 0.2 0.4 0.6 0.8 1 The scaled, 5, th moment about the mean is is / n 3 7 n 2 8 n 3 6 n 3 5 |- (1 + 4 ((1/a) ) a - 6 ((1/a) ) a + 8 ((1/a) ) a - 12 ((1/a) ) a \ 10 n 3 5 n 3 n 2 4 n - a - 44 ((1/a) ) - 72 a (1/a) + 138 a ((1/a) ) - 148 a (1/a) n 2 2 3 n n 2 n 2 + 144 ((1/a) ) a - 130 a (1/a) + 108 ((1/a) ) a + 66 ((1/a) ) 2 n n 9 n n 2 7 - 40 a (1/a) - 22 a (1/a) + 4 a (1/a) + 12 ((1/a) ) a 8 n n 2 6 n 7 n 3 4 - 38 a (1/a) + 96 ((1/a) ) a - 80 (1/a) a - 56 ((1/a) ) a n 2 5 n 6 n 3 3 n 2 4 + 162 ((1/a) ) a - 50 (1/a) a - 104 ((1/a) ) a + 180 ((1/a) ) a n 3 2 n 3 n 2 3 - 108 ((1/a) ) a - 88 ((1/a) ) a - 24 (1/a) - 2 a + 25 a + 21 a 4 7 6 5 9 8 2 / - 26 a + 19 a + 46 a + 20 a + 22 a - 25 a ) (a - 1) (a + 1) / ( / n 3 n 3 2 2 4 3 2 2 2 2 ((1/a) - 1) ((1/a) - a) (a + a + 1) (a + a + a + a + 1) (a + 1) \1/2 )| / and in Maple notation (-1/((1/a)^n-1)^3/((1/a)^n-a)^3*(1+4*((1/a)^n)^3*a^7-6*((1/a)^n)^2*a^8+8*((1/a) ^n)^3*a^6-12*((1/a)^n)^3*a^5-a^10-44*((1/a)^n)^3-72*a^5*(1/a)^n+138*a^3*((1/a)^ n)^2-148*a^4*(1/a)^n+144*((1/a)^n)^2*a^2-130*a^3*(1/a)^n+108*((1/a)^n)^2*a+66*( (1/a)^n)^2-40*a^2*(1/a)^n-22*a*(1/a)^n+4*a^9*(1/a)^n+12*((1/a)^n)^2*a^7-38*a^8* (1/a)^n+96*((1/a)^n)^2*a^6-80*(1/a)^n*a^7-56*((1/a)^n)^3*a^4+162*((1/a)^n)^2*a^ 5-50*(1/a)^n*a^6-104*((1/a)^n)^3*a^3+180*((1/a)^n)^2*a^4-108*((1/a)^n)^3*a^2-88 *((1/a)^n)^3*a-24*(1/a)^n-2*a+25*a^2+21*a^3-26*a^4+19*a^7+46*a^6+20*a^5+22*a^9-\ 25*a^8)^2/(a^2+a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^2+1)^2*(a-1)*(a+1))^(1/2) as n goes to infinity, the limiting, 5, scaled moment about the mean is / 4 3 2 2 3\1/2 | 16 (a - 1) (a + a - 5 a - 11 a - 11) (a + 1) | |- -------------------------------------------------| | 2 2 4 3 2 2 | \ (a + a + 1) (a + a + a + a + 1) / and in Maple notation (-16*(a-1)*(a^4+a^3-5*a^2-11*a-11)^2*(a+1)^3/(a^2+a+1)^2/(a^4+a^3+a^2+a+1)^2)^( 1/2) Here is a plot from a=0 to a=1 *HHHHHHHHH + HHHHHHH 40+ HHHHHH + HHHHH + HHHH + HHHH 30+ HHHH + HHHH + HHHH + HHHH + HHH 20+ HHHH + HHHH + HHHH + HHHH 10+ HHHH + HHHH + HHHHH + HHHH + HHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+--+--+--* 0 0.2 0.4 0.6 0.8 1 The scaled, 6, th moment about the mean is is n 3 13 n 2 14 n 15 n 4 11 (1 - 10 ((1/a) ) a + 10 ((1/a) ) a - 5 (1/a) a + 10 ((1/a) ) a n 3 12 n 2 13 n 4 10 + 30 ((1/a) ) a - 120 ((1/a) ) a - 35 ((1/a) ) a n 3 11 n 4 9 n 3 10 + 290 ((1/a) ) a - 165 ((1/a) ) a + 640 ((1/a) ) a n 4 8 n 4 7 n 4 6 16 15 - 345 ((1/a) ) a - 400 ((1/a) ) a - 270 ((1/a) ) a + a - 53 a 14 n n 2 12 13 n n 2 11 + 135 a (1/a) - 430 ((1/a) ) a + 20 a (1/a) - 350 ((1/a) ) a n 12 n 3 9 n 2 10 n 11 - 295 (1/a) a + 780 ((1/a) ) a + 50 ((1/a) ) a - 155 (1/a) a n 3 8 n 2 9 n 10 n 3 7 + 290 ((1/a) ) a + 580 ((1/a) ) a + 45 (1/a) a - 280 ((1/a) ) a n 2 8 n 4 5 n 3 6 n 4 4 + 340 ((1/a) ) a + 40 ((1/a) ) a - 800 ((1/a) ) a + 435 ((1/a) ) a n 3 5 n 4 3 n 4 2 - 1190 ((1/a) ) a + 615 ((1/a) ) a + 665 ((1/a) ) a n 4 10 11 13 12 14 + 530 ((1/a) ) a + 437 a + 205 a - 187 a - 28 a + 218 a n 4 n 3 5 n 3 n 2 + 265 ((1/a) ) - 530 ((1/a) ) - 345 a (1/a) + 1040 a ((1/a) ) 4 n n 2 2 3 n n 2 - 625 a (1/a) + 730 ((1/a) ) a - 665 a (1/a) + 420 ((1/a) ) a n 2 2 n n 9 n + 320 ((1/a) ) - 200 a (1/a) - 15 a (1/a) - 230 a (1/a) n 2 7 8 n n 2 6 n 7 + 600 ((1/a) ) a - 1155 a (1/a) + 1190 ((1/a) ) a - 1305 (1/a) a n 3 4 n 2 5 n 6 - 1140 ((1/a) ) a + 1970 ((1/a) ) a - 550 (1/a) a n 3 3 n 2 4 n 3 2 - 1360 ((1/a) ) a + 1750 ((1/a) ) a - 1190 ((1/a) ) a n 3 n n 4 12 2 3 - 930 ((1/a) ) a - 55 (1/a) - 3 a + 5 ((1/a) ) a + 58 a + 13 a 4 7 6 5 9 8 / n 2 - 118 a + 240 a + 427 a + 245 a + 170 a - 276 a ) / (((1/a) - 1) / n 2 2 4 3 2 2 2 2 ((1/a) - a) (a - a + 1) (a + a + a + a + 1) (a + 1) (a + a + 1) ) and in Maple notation 1/((1/a)^n-1)^2/((1/a)^n-a)^2*(1-10*((1/a)^n)^3*a^13+10*((1/a)^n)^2*a^14-5*(1/a )^n*a^15+10*((1/a)^n)^4*a^11+30*((1/a)^n)^3*a^12-120*((1/a)^n)^2*a^13-35*((1/a) ^n)^4*a^10+290*((1/a)^n)^3*a^11-165*((1/a)^n)^4*a^9+640*((1/a)^n)^3*a^10-345*(( 1/a)^n)^4*a^8-400*((1/a)^n)^4*a^7-270*((1/a)^n)^4*a^6+a^16-53*a^15+135*a^14*(1/ a)^n-430*((1/a)^n)^2*a^12+20*a^13*(1/a)^n-350*((1/a)^n)^2*a^11-295*(1/a)^n*a^12 +780*((1/a)^n)^3*a^9+50*((1/a)^n)^2*a^10-155*(1/a)^n*a^11+290*((1/a)^n)^3*a^8+ 580*((1/a)^n)^2*a^9+45*(1/a)^n*a^10-280*((1/a)^n)^3*a^7+340*((1/a)^n)^2*a^8+40* ((1/a)^n)^4*a^5-800*((1/a)^n)^3*a^6+435*((1/a)^n)^4*a^4-1190*((1/a)^n)^3*a^5+ 615*((1/a)^n)^4*a^3+665*((1/a)^n)^4*a^2+530*((1/a)^n)^4*a+437*a^10+205*a^11-187 *a^13-28*a^12+218*a^14+265*((1/a)^n)^4-530*((1/a)^n)^3-345*a^5*(1/a)^n+1040*a^3 *((1/a)^n)^2-625*a^4*(1/a)^n+730*((1/a)^n)^2*a^2-665*a^3*(1/a)^n+420*((1/a)^n)^ 2*a+320*((1/a)^n)^2-200*a^2*(1/a)^n-15*a*(1/a)^n-230*a^9*(1/a)^n+600*((1/a)^n)^ 2*a^7-1155*a^8*(1/a)^n+1190*((1/a)^n)^2*a^6-1305*(1/a)^n*a^7-1140*((1/a)^n)^3*a ^4+1970*((1/a)^n)^2*a^5-550*(1/a)^n*a^6-1360*((1/a)^n)^3*a^3+1750*((1/a)^n)^2*a ^4-1190*((1/a)^n)^3*a^2-930*((1/a)^n)^3*a-55*(1/a)^n-3*a+5*((1/a)^n)^4*a^12+58* a^2+13*a^3-118*a^4+240*a^7+427*a^6+245*a^5+170*a^9-276*a^8)/(a^2-a+1)/(a^4+a^3+ a^2+a+1)/(a^2+1)/(a^2+a+1)^2 as n goes to infinity, the limiting, 6, -th scaled moment about the mean is is 8 7 6 5 4 3 2 5 (a + a - 9 a - 26 a - 36 a - 10 a + 27 a + 53 a + 53) -------------------------------------------------------------- 2 2 2 2 (a + 1) (a - a + 1) (a + a + 1) and in Maple notation 5*(a^8+a^7-9*a^6-26*a^5-36*a^4-10*a^3+27*a^2+53*a+53)/(a^2+1)/(a^2-a+1)/(a^2+a+ 1)^2 Here is a plot from a=0 to a=1 *HHHHHHHH 250 HHHHHH + HHHHH + HHHHH + HHHH 200 HHHH + HHH + HHHH + HHHH 150 HHH + HHHH + HHH + HHH 100 HHHH + HHHH + HHHH + HHHHH 50+ HHHHH + HHHHH + HHHHHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+--+-***** 0 0.2 0.4 0.6 0.8 1 The scaled, 7, th moment about the mean is is / n 5 17 n 5 16 n 5 15 |- (-1 + 6 ((1/a) ) a + 18 ((1/a) ) a - 54 ((1/a) ) a \ n 5 14 n 2 16 n 17 - 426 ((1/a) ) a - 5430 ((1/a) ) a + 1996 (1/a) a n 3 14 n 2 15 n 16 + 4810 ((1/a) ) a - 12555 ((1/a) ) a + 2758 (1/a) a n 3 13 n 2 14 n 15 + 16740 ((1/a) ) a - 16320 ((1/a) ) a + 5214 (1/a) a n 4 11 n 3 12 n 2 13 - 7605 ((1/a) ) a + 30410 ((1/a) ) a - 22725 ((1/a) ) a n 4 10 n 3 11 n 4 9 - 21660 ((1/a) ) a + 43240 ((1/a) ) a - 34950 ((1/a) ) a n 3 10 n 4 8 n 5 6 + 54140 ((1/a) ) a - 46020 ((1/a) ) a + 15354 ((1/a) ) a n 4 7 n 5 5 n 4 6 - 53385 ((1/a) ) a + 17988 ((1/a) ) a - 55110 ((1/a) ) a n 5 4 n 5 3 n 5 2 + 17334 ((1/a) ) a + 14526 ((1/a) ) a + 10194 ((1/a) ) a n 5 20 n n 2 18 19 n + 5562 ((1/a) ) a - 368 a (1/a) + 1125 ((1/a) ) a + 740 a (1/a) n 2 17 n 18 n 3 15 16 - 750 ((1/a) ) a + 2190 (1/a) a - 2690 ((1/a) ) a - 2664 a 15 n 5 14 n n 2 12 - 2072 a + 1854 ((1/a) ) + 12682 a (1/a) - 33510 ((1/a) ) a 13 n n 2 11 n 12 + 20266 a (1/a) - 50955 ((1/a) ) a + 18910 (1/a) a n 3 9 n 2 10 n 11 + 68140 ((1/a) ) a - 61800 ((1/a) ) a + 14174 (1/a) a n 3 8 n 2 9 n 10 + 79640 ((1/a) ) a - 61455 ((1/a) ) a + 15406 (1/a) a n 3 7 n 2 8 n 4 5 + 83050 ((1/a) ) a - 52830 ((1/a) ) a - 51240 ((1/a) ) a n 3 6 n 4 4 n 3 5 + 77220 ((1/a) ) a - 45210 ((1/a) ) a + 61370 ((1/a) ) a n 4 3 n 4 2 n 4 10 - 34815 ((1/a) ) a - 23640 ((1/a) ) a - 12975 ((1/a) ) a - 3258 a 11 13 12 14 18 17 20 - 6144 a + 943 a - 4770 a + 1024 a - 139 a - 1254 a - 933 a 19 n 4 n 3 5 n + 1182 a - 4635 ((1/a) ) + 3940 ((1/a) ) + 4830 a (1/a) 3 n 2 4 n n 2 2 3 n - 12510 a ((1/a) ) + 4516 a (1/a) - 6015 ((1/a) ) a + 3478 a (1/a) n 2 n 2 2 n n - 2475 ((1/a) ) a - 1275 ((1/a) ) + 1020 a (1/a) + 80 a (1/a) 9 n n 2 7 8 n + 20702 a (1/a) - 46665 ((1/a) ) a + 22786 a (1/a) n 2 6 n 7 n 3 4 - 41520 ((1/a) ) a + 16714 (1/a) a + 43790 ((1/a) ) a n 2 5 n 6 n 3 3 - 34815 ((1/a) ) a + 8182 (1/a) a + 30730 ((1/a) ) a n 2 4 n 3 2 n 3 n - 23070 ((1/a) ) a + 18450 ((1/a) ) a + 9810 ((1/a) ) a + 118 (1/a) n 4 18 n 3 19 n 2 20 + 3 a - 15 ((1/a) ) a + 20 ((1/a) ) a - 15 ((1/a) ) a n 21 n 4 17 n 3 18 n 2 19 + 6 (1/a) a + 45 ((1/a) ) a - 270 ((1/a) ) a + 465 ((1/a) ) a n 4 16 n 3 17 n 4 15 + 720 ((1/a) ) a - 1710 ((1/a) ) a + 2565 ((1/a) ) a n 3 16 n 5 13 n 4 14 - 3430 ((1/a) ) a - 1314 ((1/a) ) a + 5190 ((1/a) ) a n 5 12 n 4 13 n 5 11 - 2508 ((1/a) ) a + 5880 ((1/a) ) a - 3294 ((1/a) ) a n 4 12 n 5 10 n 5 9 + 2010 ((1/a) ) a - 2652 ((1/a) ) a + 228 ((1/a) ) a n 5 8 n 5 7 22 21 2 3 + 4932 ((1/a) ) a + 10452 ((1/a) ) a - a + 115 a - 121 a - 78 a 4 7 6 5 9 8 2 + 253 a - 2128 a - 2440 a - 1002 a + 355 a + 1080 a ) (a + 1) / n 5 n 5 2 2 (a - 1) / (((1/a) - 1) ((1/a) - a) (a - a + 1) / 4 3 2 2 2 2 2 4 (a + a + a + a + 1) (a + 1) (a + a + 1) 6 5 4 3 2 2 \1/2 (a + a + a + a + a + a + 1) )| / and in Maple notation (-1/((1/a)^n-1)^5/((1/a)^n-a)^5*(-1+6*((1/a)^n)^5*a^17+18*((1/a)^n)^5*a^16-54*( (1/a)^n)^5*a^15-426*((1/a)^n)^5*a^14-5430*((1/a)^n)^2*a^16+1996*(1/a)^n*a^17+ 4810*((1/a)^n)^3*a^14-12555*((1/a)^n)^2*a^15+2758*(1/a)^n*a^16+16740*((1/a)^n)^ 3*a^13-16320*((1/a)^n)^2*a^14+5214*(1/a)^n*a^15-7605*((1/a)^n)^4*a^11+30410*((1 /a)^n)^3*a^12-22725*((1/a)^n)^2*a^13-21660*((1/a)^n)^4*a^10+43240*((1/a)^n)^3*a ^11-34950*((1/a)^n)^4*a^9+54140*((1/a)^n)^3*a^10-46020*((1/a)^n)^4*a^8+15354*(( 1/a)^n)^5*a^6-53385*((1/a)^n)^4*a^7+17988*((1/a)^n)^5*a^5-55110*((1/a)^n)^4*a^6 +17334*((1/a)^n)^5*a^4+14526*((1/a)^n)^5*a^3+10194*((1/a)^n)^5*a^2+5562*((1/a)^ n)^5*a-368*a^20*(1/a)^n+1125*((1/a)^n)^2*a^18+740*a^19*(1/a)^n-750*((1/a)^n)^2* a^17+2190*(1/a)^n*a^18-2690*((1/a)^n)^3*a^15-2664*a^16-2072*a^15+1854*((1/a)^n) ^5+12682*a^14*(1/a)^n-33510*((1/a)^n)^2*a^12+20266*a^13*(1/a)^n-50955*((1/a)^n) ^2*a^11+18910*(1/a)^n*a^12+68140*((1/a)^n)^3*a^9-61800*((1/a)^n)^2*a^10+14174*( 1/a)^n*a^11+79640*((1/a)^n)^3*a^8-61455*((1/a)^n)^2*a^9+15406*(1/a)^n*a^10+ 83050*((1/a)^n)^3*a^7-52830*((1/a)^n)^2*a^8-51240*((1/a)^n)^4*a^5+77220*((1/a)^ n)^3*a^6-45210*((1/a)^n)^4*a^4+61370*((1/a)^n)^3*a^5-34815*((1/a)^n)^4*a^3-\ 23640*((1/a)^n)^4*a^2-12975*((1/a)^n)^4*a-3258*a^10-6144*a^11+943*a^13-4770*a^ 12+1024*a^14-139*a^18-1254*a^17-933*a^20+1182*a^19-4635*((1/a)^n)^4+3940*((1/a) ^n)^3+4830*a^5*(1/a)^n-12510*a^3*((1/a)^n)^2+4516*a^4*(1/a)^n-6015*((1/a)^n)^2* a^2+3478*a^3*(1/a)^n-2475*((1/a)^n)^2*a-1275*((1/a)^n)^2+1020*a^2*(1/a)^n+80*a* (1/a)^n+20702*a^9*(1/a)^n-46665*((1/a)^n)^2*a^7+22786*a^8*(1/a)^n-41520*((1/a)^ n)^2*a^6+16714*(1/a)^n*a^7+43790*((1/a)^n)^3*a^4-34815*((1/a)^n)^2*a^5+8182*(1/ a)^n*a^6+30730*((1/a)^n)^3*a^3-23070*((1/a)^n)^2*a^4+18450*((1/a)^n)^3*a^2+9810 *((1/a)^n)^3*a+118*(1/a)^n+3*a-15*((1/a)^n)^4*a^18+20*((1/a)^n)^3*a^19-15*((1/a )^n)^2*a^20+6*(1/a)^n*a^21+45*((1/a)^n)^4*a^17-270*((1/a)^n)^3*a^18+465*((1/a)^ n)^2*a^19+720*((1/a)^n)^4*a^16-1710*((1/a)^n)^3*a^17+2565*((1/a)^n)^4*a^15-3430 *((1/a)^n)^3*a^16-1314*((1/a)^n)^5*a^13+5190*((1/a)^n)^4*a^14-2508*((1/a)^n)^5* a^12+5880*((1/a)^n)^4*a^13-3294*((1/a)^n)^5*a^11+2010*((1/a)^n)^4*a^12-2652*((1 /a)^n)^5*a^10+228*((1/a)^n)^5*a^9+4932*((1/a)^n)^5*a^8+10452*((1/a)^n)^5*a^7-a^ 22+115*a^21-121*a^2-78*a^3+253*a^4-2128*a^7-2440*a^6-1002*a^5+355*a^9+1080*a^8) ^2*(a+1)*(a-1)/(a^2-a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^2+1)^2/(a^2+a+1)^4/(a^6+a^5+a ^4+a^3+a^2+a+1)^2)^(1/2) as n goes to infinity, the limiting, 7, scaled moment about the mean is / 12 11 10 9 8 7 6 5 |- 36 (a - 1) (a + 2 a - 12 a - 62 a - 148 a - 199 a - 130 a + 109 a \ 4 3 2 2 3 / 2 2 + 468 a + 722 a + 772 a + 618 a + 309) (a + 1) / ((a + a + 1) / 4 3 2 2 6 5 4 3 2 2 2 2 \1/2 (a + a + a + a + 1) (a + a + a + a + a + a + 1) (a + 1) )| / and in Maple notation (-36*(a-1)*(a^12+2*a^11-12*a^10-62*a^9-148*a^8-199*a^7-130*a^6+109*a^5+468*a^4+ 722*a^3+772*a^2+618*a+309)^2*(a+1)^3/(a^2+a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^6+a^5+a ^4+a^3+a^2+a+1)^2/(a^2+1)^2)^(1/2) Here is a plot from a=0 to a=1 *HHHHHHH 1800 HHHHHH + HHHH 1600 HHHH + HHHH 1400 HHHH + HHHH 1200 HHH + HHH 1000 HHH + HHH 800 HHH + HHHH + HHH 600 HHH + HHHH 400 HHHH + HHHHH 200 HHHHHH + HHHHHHHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+--+--**** 0 0.2 0.4 0.6 0.8 1 The scaled, 8, th moment about the mean is is n 6 20 n 6 19 n 6 18 - (-1 - 2779 ((1/a) ) a - 5824 ((1/a) ) a - 8071 ((1/a) ) a n 6 17 n 6 16 n 6 24 - 5607 ((1/a) ) a + 6258 ((1/a) ) a + 7 ((1/a) ) a n 5 25 n 6 23 n 5 24 - 21 ((1/a) ) a + 21 ((1/a) ) a + 84 ((1/a) ) a n 6 22 n 5 23 n 6 21 - 105 ((1/a) ) a + 1428 ((1/a) ) a - 826 ((1/a) ) a n 5 22 n 6 8 n 4 26 + 5817 ((1/a) ) a - 22302 ((1/a) ) a + 35 ((1/a) ) a n 3 27 n 2 28 n 29 n 4 25 - 35 ((1/a) ) a + 21 ((1/a) ) a - 7 (1/a) a - 560 ((1/a) ) a n 3 26 n 2 27 n 28 + 1260 ((1/a) ) a - 1484 ((1/a) ) a + 924 (1/a) a n 4 24 n 3 25 n 2 26 - 4305 ((1/a) ) a + 4585 ((1/a) ) a + 1281 ((1/a) ) a n 4 23 n 3 24 n 5 21 - 10430 ((1/a) ) a - 1085 ((1/a) ) a + 13167 ((1/a) ) a n 4 22 n 3 23 n 5 20 - 8610 ((1/a) ) a - 29995 ((1/a) ) a + 15561 ((1/a) ) a n 4 21 n 5 19 n 4 20 + 27195 ((1/a) ) a - 1323 ((1/a) ) a + 109410 ((1/a) ) a n 5 18 n 4 19 n 5 17 - 51702 ((1/a) ) a + 228830 ((1/a) ) a - 138705 ((1/a) ) a n 6 15 n 5 16 n 6 14 + 29106 ((1/a) ) a - 238308 ((1/a) ) a + 59906 ((1/a) ) a n 5 15 n 6 13 n 5 14 - 319956 ((1/a) ) a + 90930 ((1/a) ) a - 355404 ((1/a) ) a n 6 12 n 6 11 n 6 10 + 111048 ((1/a) ) a + 109410 ((1/a) ) a + 83986 ((1/a) ) a n 6 9 23 n 2 16 n 17 + 37506 ((1/a) ) a - 6459 a - 198408 ((1/a) ) a + 39487 (1/a) a n 3 14 n 2 15 n 16 + 54250 ((1/a) ) a - 297864 ((1/a) ) a + 27622 (1/a) a n 3 13 n 2 14 n 15 + 354970 ((1/a) ) a - 309344 ((1/a) ) a + 87906 (1/a) a n 4 11 n 3 12 n 2 13 - 283850 ((1/a) ) a + 619290 ((1/a) ) a - 313208 ((1/a) ) a n 4 10 n 3 11 n 4 9 - 557970 ((1/a) ) a + 747880 ((1/a) ) a - 776895 ((1/a) ) a n 3 10 n 4 8 n 5 6 + 802445 ((1/a) ) a - 908320 ((1/a) ) a + 495117 ((1/a) ) a n 4 7 n 5 5 n 4 6 - 935550 ((1/a) ) a + 491841 ((1/a) ) a - 826350 ((1/a) ) a n 5 4 n 5 3 n 5 2 + 428967 ((1/a) ) a + 337617 ((1/a) ) a + 229908 ((1/a) ) a n 5 20 n n 2 18 + 126084 ((1/a) ) a + 24241 a (1/a) + 106792 ((1/a) ) a 19 n n 2 17 n 18 + 105532 a (1/a) - 19621 ((1/a) ) a + 100058 (1/a) a n 3 15 n 6 16 15 - 161350 ((1/a) ) a - 14833 ((1/a) ) - 68404 a - 21868 a n 5 14 n n 2 12 + 44499 ((1/a) ) + 202202 a (1/a) - 383880 ((1/a) ) a 13 n n 2 11 n 12 + 267470 a (1/a) - 514437 ((1/a) ) a + 211379 (1/a) a n 3 9 n 2 10 n 11 + 828730 ((1/a) ) a - 577024 ((1/a) ) a + 110698 (1/a) a n 3 8 n 2 9 n 10 + 821380 ((1/a) ) a - 535262 ((1/a) ) a + 79464 (1/a) a n 3 7 n 2 8 n 4 5 + 782285 ((1/a) ) a - 399245 ((1/a) ) a - 671685 ((1/a) ) a n 3 6 n 4 4 n 3 5 + 674380 ((1/a) ) a - 530810 ((1/a) ) a + 501270 ((1/a) ) a n 4 3 n 4 2 n 4 - 374010 ((1/a) ) a - 235865 ((1/a) ) a - 126700 ((1/a) ) a 10 11 13 12 14 18 - 34575 a - 54254 a + 20829 a - 38868 a + 23240 a - 28150 a 17 20 19 n 4 n 3 - 66693 a + 21681 a + 35564 a - 48825 ((1/a) ) + 23485 ((1/a) ) 5 n 3 n 2 4 n + 13230 a (1/a) - 62909 a ((1/a) ) + 9786 a (1/a) n 2 2 3 n n 2 n 2 - 25599 ((1/a) ) a + 11375 a (1/a) - 7140 ((1/a) ) a - 4571 ((1/a) ) 2 n n 9 n n 2 7 + 3829 a (1/a) + 28 a (1/a) + 99757 a (1/a) - 298865 ((1/a) ) a 8 n n 2 6 n 7 + 121989 a (1/a) - 221655 ((1/a) ) a + 98112 (1/a) a n 3 4 n 2 5 n 6 + 325605 ((1/a) ) a - 177331 ((1/a) ) a + 42364 (1/a) a n 3 3 n 2 4 n 3 2 + 208915 ((1/a) ) a - 115318 ((1/a) ) a + 109305 ((1/a) ) a n 3 n 27 n + 52220 ((1/a) ) a + 245 (1/a) + 4 a - 6475 a (1/a) n 2 25 26 n n 2 24 + 16443 ((1/a) ) a - 2541 a (1/a) + 32074 ((1/a) ) a n 25 n 3 22 n 2 23 + 6258 (1/a) a - 97370 ((1/a) ) a + 51149 ((1/a) ) a n 24 n 3 21 n 2 22 - 4326 (1/a) a - 164500 ((1/a) ) a + 36449 ((1/a) ) a n 23 n 3 20 n 2 21 - 5208 (1/a) a - 212835 ((1/a) ) a + 45759 ((1/a) ) a n 22 n 4 18 n 3 19 - 23436 (1/a) a + 349020 ((1/a) ) a - 224700 ((1/a) ) a n 2 20 n 21 n 4 17 + 70147 ((1/a) ) a - 34363 (1/a) a + 409605 ((1/a) ) a n 3 18 n 2 19 n 4 16 - 221270 ((1/a) ) a + 134050 ((1/a) ) a + 415870 ((1/a) ) a n 3 17 n 4 15 n 3 16 - 254275 ((1/a) ) a + 375830 ((1/a) ) a - 252840 ((1/a) ) a n 5 13 n 4 14 n 5 12 - 324198 ((1/a) ) a + 289380 ((1/a) ) a - 223398 ((1/a) ) a n 4 13 n 5 11 n 4 12 + 149940 ((1/a) ) a - 89964 ((1/a) ) a - 23380 ((1/a) ) a n 5 10 n 5 9 n 6 7 + 68124 ((1/a) ) a + 233772 ((1/a) ) a - 82047 ((1/a) ) a n 5 8 n 6 6 n 5 7 + 374535 ((1/a) ) a - 125951 ((1/a) ) a + 464058 ((1/a) ) a n 6 5 n 6 4 n 6 3 - 143024 ((1/a) ) a - 139139 ((1/a) ) a - 116186 ((1/a) ) a n 6 2 n 6 22 21 2 - 81585 ((1/a) ) a - 44499 ((1/a) ) a - 28168 a - 22025 a - 251 a 3 4 7 6 5 9 8 - 23 a + 738 a - 2543 a - 7673 a - 4232 a - 3007 a + 12474 a 28 26 27 30 29 24 25 + 3849 a + 10056 a - 12157 a + a - 242 a + 3011 a + 3546 a ) / n 3 n 3 2 2 2 2 / (((1/a) - 1) ((1/a) - a) (a + a + 1) (a + 1) / 4 3 2 2 6 5 4 3 2 (a + a + a + a + 1) (a - a + 1) (a + a + a + a + a + a + 1) 4 (a + 1)) and in Maple notation -1/((1/a)^n-1)^3/((1/a)^n-a)^3*(-1-2779*((1/a)^n)^6*a^20-5824*((1/a)^n)^6*a^19-\ 8071*((1/a)^n)^6*a^18-5607*((1/a)^n)^6*a^17+6258*((1/a)^n)^6*a^16+7*((1/a)^n)^6 *a^24-21*((1/a)^n)^5*a^25+21*((1/a)^n)^6*a^23+84*((1/a)^n)^5*a^24-105*((1/a)^n) ^6*a^22+1428*((1/a)^n)^5*a^23-826*((1/a)^n)^6*a^21+5817*((1/a)^n)^5*a^22-22302* ((1/a)^n)^6*a^8+35*((1/a)^n)^4*a^26-35*((1/a)^n)^3*a^27+21*((1/a)^n)^2*a^28-7*( 1/a)^n*a^29-560*((1/a)^n)^4*a^25+1260*((1/a)^n)^3*a^26-1484*((1/a)^n)^2*a^27+ 924*(1/a)^n*a^28-4305*((1/a)^n)^4*a^24+4585*((1/a)^n)^3*a^25+1281*((1/a)^n)^2*a ^26-10430*((1/a)^n)^4*a^23-1085*((1/a)^n)^3*a^24+13167*((1/a)^n)^5*a^21-8610*(( 1/a)^n)^4*a^22-29995*((1/a)^n)^3*a^23+15561*((1/a)^n)^5*a^20+27195*((1/a)^n)^4* a^21-1323*((1/a)^n)^5*a^19+109410*((1/a)^n)^4*a^20-51702*((1/a)^n)^5*a^18+ 228830*((1/a)^n)^4*a^19-138705*((1/a)^n)^5*a^17+29106*((1/a)^n)^6*a^15-238308*( (1/a)^n)^5*a^16+59906*((1/a)^n)^6*a^14-319956*((1/a)^n)^5*a^15+90930*((1/a)^n)^ 6*a^13-355404*((1/a)^n)^5*a^14+111048*((1/a)^n)^6*a^12+109410*((1/a)^n)^6*a^11+ 83986*((1/a)^n)^6*a^10+37506*((1/a)^n)^6*a^9-6459*a^23-198408*((1/a)^n)^2*a^16+ 39487*(1/a)^n*a^17+54250*((1/a)^n)^3*a^14-297864*((1/a)^n)^2*a^15+27622*(1/a)^n *a^16+354970*((1/a)^n)^3*a^13-309344*((1/a)^n)^2*a^14+87906*(1/a)^n*a^15-283850 *((1/a)^n)^4*a^11+619290*((1/a)^n)^3*a^12-313208*((1/a)^n)^2*a^13-557970*((1/a) ^n)^4*a^10+747880*((1/a)^n)^3*a^11-776895*((1/a)^n)^4*a^9+802445*((1/a)^n)^3*a^ 10-908320*((1/a)^n)^4*a^8+495117*((1/a)^n)^5*a^6-935550*((1/a)^n)^4*a^7+491841* ((1/a)^n)^5*a^5-826350*((1/a)^n)^4*a^6+428967*((1/a)^n)^5*a^4+337617*((1/a)^n)^ 5*a^3+229908*((1/a)^n)^5*a^2+126084*((1/a)^n)^5*a+24241*a^20*(1/a)^n+106792*((1 /a)^n)^2*a^18+105532*a^19*(1/a)^n-19621*((1/a)^n)^2*a^17+100058*(1/a)^n*a^18-\ 161350*((1/a)^n)^3*a^15-14833*((1/a)^n)^6-68404*a^16-21868*a^15+44499*((1/a)^n) ^5+202202*a^14*(1/a)^n-383880*((1/a)^n)^2*a^12+267470*a^13*(1/a)^n-514437*((1/a )^n)^2*a^11+211379*(1/a)^n*a^12+828730*((1/a)^n)^3*a^9-577024*((1/a)^n)^2*a^10+ 110698*(1/a)^n*a^11+821380*((1/a)^n)^3*a^8-535262*((1/a)^n)^2*a^9+79464*(1/a)^n *a^10+782285*((1/a)^n)^3*a^7-399245*((1/a)^n)^2*a^8-671685*((1/a)^n)^4*a^5+ 674380*((1/a)^n)^3*a^6-530810*((1/a)^n)^4*a^4+501270*((1/a)^n)^3*a^5-374010*((1 /a)^n)^4*a^3-235865*((1/a)^n)^4*a^2-126700*((1/a)^n)^4*a-34575*a^10-54254*a^11+ 20829*a^13-38868*a^12+23240*a^14-28150*a^18-66693*a^17+21681*a^20+35564*a^19-\ 48825*((1/a)^n)^4+23485*((1/a)^n)^3+13230*a^5*(1/a)^n-62909*a^3*((1/a)^n)^2+ 9786*a^4*(1/a)^n-25599*((1/a)^n)^2*a^2+11375*a^3*(1/a)^n-7140*((1/a)^n)^2*a-\ 4571*((1/a)^n)^2+3829*a^2*(1/a)^n+28*a*(1/a)^n+99757*a^9*(1/a)^n-298865*((1/a)^ n)^2*a^7+121989*a^8*(1/a)^n-221655*((1/a)^n)^2*a^6+98112*(1/a)^n*a^7+325605*((1 /a)^n)^3*a^4-177331*((1/a)^n)^2*a^5+42364*(1/a)^n*a^6+208915*((1/a)^n)^3*a^3-\ 115318*((1/a)^n)^2*a^4+109305*((1/a)^n)^3*a^2+52220*((1/a)^n)^3*a+245*(1/a)^n+4 *a-6475*a^27*(1/a)^n+16443*((1/a)^n)^2*a^25-2541*a^26*(1/a)^n+32074*((1/a)^n)^2 *a^24+6258*(1/a)^n*a^25-97370*((1/a)^n)^3*a^22+51149*((1/a)^n)^2*a^23-4326*(1/a )^n*a^24-164500*((1/a)^n)^3*a^21+36449*((1/a)^n)^2*a^22-5208*(1/a)^n*a^23-\ 212835*((1/a)^n)^3*a^20+45759*((1/a)^n)^2*a^21-23436*(1/a)^n*a^22+349020*((1/a) ^n)^4*a^18-224700*((1/a)^n)^3*a^19+70147*((1/a)^n)^2*a^20-34363*(1/a)^n*a^21+ 409605*((1/a)^n)^4*a^17-221270*((1/a)^n)^3*a^18+134050*((1/a)^n)^2*a^19+415870* ((1/a)^n)^4*a^16-254275*((1/a)^n)^3*a^17+375830*((1/a)^n)^4*a^15-252840*((1/a)^ n)^3*a^16-324198*((1/a)^n)^5*a^13+289380*((1/a)^n)^4*a^14-223398*((1/a)^n)^5*a^ 12+149940*((1/a)^n)^4*a^13-89964*((1/a)^n)^5*a^11-23380*((1/a)^n)^4*a^12+68124* ((1/a)^n)^5*a^10+233772*((1/a)^n)^5*a^9-82047*((1/a)^n)^6*a^7+374535*((1/a)^n)^ 5*a^8-125951*((1/a)^n)^6*a^6+464058*((1/a)^n)^5*a^7-143024*((1/a)^n)^6*a^5-\ 139139*((1/a)^n)^6*a^4-116186*((1/a)^n)^6*a^3-81585*((1/a)^n)^6*a^2-44499*((1/a )^n)^6*a-28168*a^22-22025*a^21-251*a^2-23*a^3+738*a^4-2543*a^7-7673*a^6-4232*a^ 5-3007*a^9+12474*a^8+3849*a^28+10056*a^26-12157*a^27+a^30-242*a^29+3011*a^24+ 3546*a^25)/(a^2+a+1)^2/(a^2+1)^2/(a^4+a^3+a^2+a+1)/(a^2-a+1)/(a^6+a^5+a^4+a^3+a ^2+a+1)/(a^4+1) as n goes to infinity, the limiting, 8, -th scaled moment about the mean is is 18 17 16 15 14 13 12 11 - 7 (a + 2 a - 18 a - 103 a - 279 a - 435 a - 321 a + 353 a 10 9 8 7 6 5 4 + 1697 a + 3246 a + 4297 a + 4153 a + 2439 a - 555 a - 3279 a 3 2 / 4 3 2 - 4943 a - 5298 a - 4238 a - 2119) / ((a + a + a + a + 1) / 2 4 2 2 2 2 (a - a + 1) (a + 1) (a + 1) (a + a + 1) ) and in Maple notation -7*(a^18+2*a^17-18*a^16-103*a^15-279*a^14-435*a^13-321*a^12+353*a^11+1697*a^10+ 3246*a^9+4297*a^8+4153*a^7+2439*a^6-555*a^5-3279*a^4-4943*a^3-5298*a^2-4238*a-\ 2119)/(a^4+a^3+a^2+a+1)/(a^2-a+1)/(a^4+1)/(a^2+1)^2/(a^2+a+1)^2 Here is a plot from a=0 to a=1 *HHHHHHH 14000 HHHHH + HHHH + HHHH 12000 HHH + HHH + HHH 10000 HHHH + HHH 8000 HHH + HHH + HHH 6000 HHH + HHHH + HHH 4000 HHHH + HHHH 2000 HHHHH + HHHHHH + HHHHHHHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+--+********* 0 0.2 0.4 0.6 0.8 1 The scaled, 9, th moment about the mean is is / n 6 32 n 5 33 n 6 31 |- (1 - 28 ((1/a) ) a + 56 ((1/a) ) a + 112 ((1/a) ) a \ n 5 32 n 6 30 n 5 31 - 980 ((1/a) ) a + 2660 ((1/a) ) a - 10304 ((1/a) ) a n 6 29 n 7 27 n 6 28 + 14364 ((1/a) ) a - 6872 ((1/a) ) a + 43960 ((1/a) ) a n 7 26 n 6 27 n 7 25 - 19048 ((1/a) ) a + 82152 ((1/a) ) a - 37240 ((1/a) ) a n 6 26 n 7 24 n 6 25 + 69748 ((1/a) ) a - 49144 ((1/a) ) a - 126560 ((1/a) ) a n 7 23 n 7 22 n 7 21 - 23920 ((1/a) ) a + 85296 ((1/a) ) a + 326440 ((1/a) ) a n 7 20 n 7 19 n 7 18 + 723216 ((1/a) ) a + 1246472 ((1/a) ) a + 1796000 ((1/a) ) a n 7 31 n 7 30 n 7 29 + 8 ((1/a) ) a + 32 ((1/a) ) a - 144 ((1/a) ) a n 7 28 n 6 20 n 6 19 - 1584 ((1/a) ) a - 6914572 ((1/a) ) a - 7644560 ((1/a) ) a n 7 17 n 6 18 n 7 16 + 2217336 ((1/a) ) a - 6939016 ((1/a) ) a + 2335568 ((1/a) ) a n 6 17 n 7 15 n 6 16 - 4491340 ((1/a) ) a + 1999120 ((1/a) ) a - 272160 ((1/a) ) a n 7 14 n 7 13 n 7 12 + 1129368 ((1/a) ) a - 233888 ((1/a) ) a - 1919096 ((1/a) ) a n 7 11 n 7 10 n 7 9 - 3688464 ((1/a) ) a - 5271544 ((1/a) ) a - 6408912 ((1/a) ) a n 4 34 n 3 35 n 2 36 n 37 - 70 ((1/a) ) a + 56 ((1/a) ) a - 28 ((1/a) ) a + 8 (1/a) a n 4 33 n 3 34 n 2 35 + 2800 ((1/a) ) a - 4424 ((1/a) ) a + 4144 ((1/a) ) a n 36 n 4 32 n 3 33 - 2174 (1/a) a + 16660 ((1/a) ) a - 5600 ((1/a) ) a n 2 34 n 5 30 n 4 31 - 20552 ((1/a) ) a - 37072 ((1/a) ) a + 19810 ((1/a) ) a n 3 32 n 5 29 n 4 30 + 61768 ((1/a) ) a - 63224 ((1/a) ) a - 87570 ((1/a) ) a n 3 31 n 5 28 n 4 29 + 269360 ((1/a) ) a + 27972 ((1/a) ) a - 539910 ((1/a) ) a n 5 27 n 4 28 n 5 26 + 471884 ((1/a) ) a - 1507450 ((1/a) ) a + 1554924 ((1/a) ) a n 4 27 n 6 24 n 5 25 - 2942240 ((1/a) ) a - 698600 ((1/a) ) a + 3430980 ((1/a) ) a n 6 23 n 5 24 n 6 22 - 1795108 ((1/a) ) a + 5822656 ((1/a) ) a - 3406424 ((1/a) ) a n 5 23 n 6 21 n 5 22 + 8069712 ((1/a) ) a - 5281556 ((1/a) ) a + 9299668 ((1/a) ) a n 7 7 n 6 8 n 7 6 - 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21 n 5 19 n 4 20 + 6952610 ((1/a) ) a + 280168 ((1/a) ) a + 14513590 ((1/a) ) a n 5 18 n 4 19 n 5 17 - 7276388 ((1/a) ) a + 25058670 ((1/a) ) a - 16899960 ((1/a) ) a n 6 15 n 5 16 n 6 14 + 5418196 ((1/a) ) a - 28151088 ((1/a) ) a + 11931640 ((1/a) ) a n 5 15 n 6 13 n 5 14 - 40090036 ((1/a) ) a + 18422096 ((1/a) ) a - 51566200 ((1/a) ) a n 6 12 n 6 11 n 6 10 + 24267796 ((1/a) ) a + 28794668 ((1/a) ) a + 31411688 ((1/a) ) a n 7 8 n 6 9 32 35 - 6915728 ((1/a) ) a + 31846108 ((1/a) ) a + 16071 a + 68923 a 23 38 34 n 2 16 + 978807 a - a - 84508 a + 33217520 ((1/a) ) a n 17 n 3 14 n 2 15 - 5924806 (1/a) a - 67815272 ((1/a) ) a + 31085936 ((1/a) ) a n 16 n 3 13 n 2 14 - 6229376 (1/a) a - 66362072 ((1/a) ) a + 27553708 ((1/a) ) a n 15 n 4 11 n 3 12 - 6985168 (1/a) a + 81974340 ((1/a) ) a - 60993464 ((1/a) ) a n 2 13 n 4 10 n 3 11 + 24133424 ((1/a) ) a + 74108020 ((1/a) ) a - 52780952 ((1/a) ) a n 4 9 n 3 10 n 4 8 + 63778470 ((1/a) ) a - 43482376 ((1/a) ) a + 51978430 ((1/a) ) a n 5 6 n 4 7 n 5 5 - 34377476 ((1/a) ) a + 39691400 ((1/a) ) a - 25026372 ((1/a) ) a n 4 6 n 5 4 n 5 3 + 28143850 ((1/a) ) a - 16767856 ((1/a) ) a - 10200176 ((1/a) ) a n 5 2 n 5 20 n - 5398120 ((1/a) ) a - 2302804 ((1/a) ) a - 8723858 a (1/a) n 2 18 19 n n 2 17 + 29099952 ((1/a) ) a - 8422342 a (1/a) + 32531828 ((1/a) ) a n 18 n 3 15 n 6 - 6930086 (1/a) a - 66029712 ((1/a) ) a + 467236 ((1/a) ) 16 15 n 5 14 n + 730357 a + 40353 a - 628544 ((1/a) ) - 7108396 a (1/a) n 2 12 13 n n 2 11 + 21236684 ((1/a) ) a - 6025374 a (1/a) + 18409160 ((1/a) ) a n 12 n 3 9 n 2 10 - 4109970 (1/a) a - 34353368 ((1/a) ) a + 14931000 ((1/a) ) a n 11 n 3 8 n 2 9 - 2480364 (1/a) a - 26145504 ((1/a) ) a + 11036760 ((1/a) ) a n 10 n 3 7 n 2 8 - 1753690 (1/a) a - 19057192 ((1/a) ) a + 7366968 ((1/a) ) a n 4 5 n 3 6 n 4 4 + 18749430 ((1/a) ) a - 12952688 ((1/a) ) a + 11722410 ((1/a) ) a n 3 5 n 4 3 n 4 2 - 8003240 ((1/a) ) a + 6610310 ((1/a) ) a + 3269420 ((1/a) ) a n 4 10 11 13 12 + 1365560 ((1/a) ) a + 247661 a + 502394 a + 149314 a + 547852 a 14 18 17 20 19 - 156579 a + 1689325 a + 1491577 a - 35497 a + 854088 a n 4 n 3 5 n + 403270 ((1/a) ) - 123368 ((1/a) ) - 114736 a (1/a) 3 n 2 4 n n 2 2 + 452228 a ((1/a) ) - 59948 a (1/a) + 148820 ((1/a) ) a 3 n n 2 n 2 2 n - 47680 a (1/a) + 33656 ((1/a) ) a + 15400 ((1/a) ) - 13900 a (1/a) n 9 n n 2 7 - 286 a (1/a) - 1520898 a (1/a) + 4714220 ((1/a) ) a 8 n n 2 6 n 7 - 1297422 a (1/a) + 2884952 ((1/a) ) a - 845082 (1/a) a n 3 4 n 2 5 n 6 - 4476080 ((1/a) ) a + 1770440 ((1/a) ) a - 352842 (1/a) a n 3 3 n 2 4 n 3 2 - 2314816 ((1/a) ) a + 960652 ((1/a) ) a - 976528 ((1/a) ) a n 3 n 27 n - 364336 ((1/a) ) a - 500 (1/a) - 4 a - 1803302 a (1/a) n 2 25 26 n n 2 24 + 4696972 ((1/a) ) a - 2545260 a (1/a) + 8597848 ((1/a) ) a n 25 n 3 22 n 2 23 - 2476998 (1/a) a - 14103208 ((1/a) ) a + 12532828 ((1/a) ) a n 24 n 3 21 n 2 22 - 2477178 (1/a) a - 23084488 ((1/a) ) a + 15814988 ((1/a) ) a n 23 n 3 20 n 2 21 - 3363464 (1/a) a - 33542880 ((1/a) ) a + 18252136 ((1/a) ) a n 22 n 4 18 n 3 19 - 5110352 (1/a) a + 38291540 ((1/a) ) a - 43372112 ((1/a) ) a n 2 20 n 21 n 4 17 + 20956180 ((1/a) ) a - 7277980 (1/a) a + 52970400 ((1/a) ) a n 3 18 n 2 19 n 4 16 - 51400048 ((1/a) ) a + 24597832 ((1/a) ) a + 66931060 ((1/a) ) a n 3 17 n 4 15 n 3 16 - 57512168 ((1/a) ) a + 78357090 ((1/a) ) a - 62347936 ((1/a) ) a n 5 13 n 4 14 n 5 12 - 61529188 ((1/a) ) a + 85399370 ((1/a) ) a - 68284776 ((1/a) ) a n 4 13 n 5 11 n 4 12 + 87897670 ((1/a) ) a - 70861756 ((1/a) ) a + 86539530 ((1/a) ) a n 5 10 n 5 9 n 6 7 - 69190296 ((1/a) ) a - 63630952 ((1/a) ) a + 26858720 ((1/a) ) a n 5 8 n 6 6 n 5 7 - 54993204 ((1/a) ) a + 22430324 ((1/a) ) a - 44650452 ((1/a) ) a n 6 5 n 6 4 n 6 3 + 17327016 ((1/a) ) a + 12290936 ((1/a) ) a + 7798140 ((1/a) ) a n 6 2 n 6 22 21 + 4238500 ((1/a) ) a + 1802192 ((1/a) ) a + 361427 a - 223669 a 2 3 4 7 6 5 9 + 506 a + 281 a - 1496 a + 26845 a + 34497 a + 14021 a + 23898 a 8 n 7 28 26 27 - 34556 a - 133496 ((1/a) ) - 43841 a + 52136 a - 314366 a 33 31 36 37 30 29 - 3617 a + 65315 a - 13418 a + 496 a + 214256 a + 263958 a 24 25 2 / n 7 + 1193247 a + 869546 a ) (a - 1) (a + 1) / (((1/a) - 1) / n 7 2 2 4 3 2 2 6 3 2 ((1/a) - a) (a - a + 1) (a + a + a + a + 1) (a + a + 1) 4 2 6 5 4 3 2 2 2 4 2 6 \1/2 (a + 1) (a + a + a + a + a + a + 1) (a + 1) (a + a + 1) )| / and in Maple notation (-1/((1/a)^n-1)^7/((1/a)^n-a)^7*(1-28*((1/a)^n)^6*a^32+56*((1/a)^n)^5*a^33+112* ((1/a)^n)^6*a^31-980*((1/a)^n)^5*a^32+2660*((1/a)^n)^6*a^30-10304*((1/a)^n)^5*a ^31+14364*((1/a)^n)^6*a^29-6872*((1/a)^n)^7*a^27+43960*((1/a)^n)^6*a^28-19048*( (1/a)^n)^7*a^26+82152*((1/a)^n)^6*a^27-37240*((1/a)^n)^7*a^25+69748*((1/a)^n)^6 *a^26-49144*((1/a)^n)^7*a^24-126560*((1/a)^n)^6*a^25-23920*((1/a)^n)^7*a^23+ 85296*((1/a)^n)^7*a^22+326440*((1/a)^n)^7*a^21+723216*((1/a)^n)^7*a^20+1246472* ((1/a)^n)^7*a^19+1796000*((1/a)^n)^7*a^18+8*((1/a)^n)^7*a^31+32*((1/a)^n)^7*a^ 30-144*((1/a)^n)^7*a^29-1584*((1/a)^n)^7*a^28-6914572*((1/a)^n)^6*a^20-7644560* ((1/a)^n)^6*a^19+2217336*((1/a)^n)^7*a^17-6939016*((1/a)^n)^6*a^18+2335568*((1/ a)^n)^7*a^16-4491340*((1/a)^n)^6*a^17+1999120*((1/a)^n)^7*a^15-272160*((1/a)^n) ^6*a^16+1129368*((1/a)^n)^7*a^14-233888*((1/a)^n)^7*a^13-1919096*((1/a)^n)^7*a^ 12-3688464*((1/a)^n)^7*a^11-5271544*((1/a)^n)^7*a^10-6408912*((1/a)^n)^7*a^9-70 *((1/a)^n)^4*a^34+56*((1/a)^n)^3*a^35-28*((1/a)^n)^2*a^36+8*(1/a)^n*a^37+2800*( (1/a)^n)^4*a^33-4424*((1/a)^n)^3*a^34+4144*((1/a)^n)^2*a^35-2174*(1/a)^n*a^36+ 16660*((1/a)^n)^4*a^32-5600*((1/a)^n)^3*a^33-20552*((1/a)^n)^2*a^34-37072*((1/a )^n)^5*a^30+19810*((1/a)^n)^4*a^31+61768*((1/a)^n)^3*a^32-63224*((1/a)^n)^5*a^ 29-87570*((1/a)^n)^4*a^30+269360*((1/a)^n)^3*a^31+27972*((1/a)^n)^5*a^28-539910 *((1/a)^n)^4*a^29+471884*((1/a)^n)^5*a^27-1507450*((1/a)^n)^4*a^28+1554924*((1/ a)^n)^5*a^26-2942240*((1/a)^n)^4*a^27-698600*((1/a)^n)^6*a^24+3430980*((1/a)^n) ^5*a^25-1795108*((1/a)^n)^6*a^23+5822656*((1/a)^n)^5*a^24-3406424*((1/a)^n)^6*a ^22+8069712*((1/a)^n)^5*a^23-5281556*((1/a)^n)^6*a^21+9299668*((1/a)^n)^5*a^22-\ 6738008*((1/a)^n)^7*a^7+30191672*((1/a)^n)^6*a^8-5977496*((1/a)^n)^7*a^6-\ 4855208*((1/a)^n)^7*a^5-3565336*((1/a)^n)^7*a^4-2313936*((1/a)^n)^7*a^3-1268208 *((1/a)^n)^7*a^2-533984*((1/a)^n)^7*a+30208*a^35*(1/a)^n-81536*((1/a)^n)^2*a^33 -29024*a^34*(1/a)^n-95536*((1/a)^n)^2*a^32-78892*(1/a)^n*a^33+703808*((1/a)^n)^ 3*a^30-115304*((1/a)^n)^2*a^31+2596*(1/a)^n*a^32+1105160*((1/a)^n)^3*a^29+ 126952*((1/a)^n)^2*a^30-78246*(1/a)^n*a^31+1216264*((1/a)^n)^3*a^28+432880*((1/ a)^n)^2*a^29-151890*(1/a)^n*a^30-4400830*((1/a)^n)^4*a^26+671664*((1/a)^n)^3*a^ 27+868392*((1/a)^n)^2*a^28-212598*(1/a)^n*a^29-5057430*((1/a)^n)^4*a^25-481264* ((1/a)^n)^3*a^26+1125516*((1/a)^n)^2*a^27-734730*(1/a)^n*a^28-4432540*((1/a)^n) ^4*a^24-1931888*((1/a)^n)^3*a^25+2221380*((1/a)^n)^2*a^26-2286900*((1/a)^n)^4*a ^23-4160800*((1/a)^n)^3*a^24+8667624*((1/a)^n)^5*a^21+1441230*((1/a)^n)^4*a^22-\ 7898296*((1/a)^n)^3*a^23+5621980*((1/a)^n)^5*a^20+6952610*((1/a)^n)^4*a^21+ 280168*((1/a)^n)^5*a^19+14513590*((1/a)^n)^4*a^20-7276388*((1/a)^n)^5*a^18+ 25058670*((1/a)^n)^4*a^19-16899960*((1/a)^n)^5*a^17+5418196*((1/a)^n)^6*a^15-\ 28151088*((1/a)^n)^5*a^16+11931640*((1/a)^n)^6*a^14-40090036*((1/a)^n)^5*a^15+ 18422096*((1/a)^n)^6*a^13-51566200*((1/a)^n)^5*a^14+24267796*((1/a)^n)^6*a^12+ 28794668*((1/a)^n)^6*a^11+31411688*((1/a)^n)^6*a^10-6915728*((1/a)^n)^7*a^8+ 31846108*((1/a)^n)^6*a^9+16071*a^32+68923*a^35+978807*a^23-a^38-84508*a^34+ 33217520*((1/a)^n)^2*a^16-5924806*(1/a)^n*a^17-67815272*((1/a)^n)^3*a^14+ 31085936*((1/a)^n)^2*a^15-6229376*(1/a)^n*a^16-66362072*((1/a)^n)^3*a^13+ 27553708*((1/a)^n)^2*a^14-6985168*(1/a)^n*a^15+81974340*((1/a)^n)^4*a^11-\ 60993464*((1/a)^n)^3*a^12+24133424*((1/a)^n)^2*a^13+74108020*((1/a)^n)^4*a^10-\ 52780952*((1/a)^n)^3*a^11+63778470*((1/a)^n)^4*a^9-43482376*((1/a)^n)^3*a^10+ 51978430*((1/a)^n)^4*a^8-34377476*((1/a)^n)^5*a^6+39691400*((1/a)^n)^4*a^7-\ 25026372*((1/a)^n)^5*a^5+28143850*((1/a)^n)^4*a^6-16767856*((1/a)^n)^5*a^4-\ 10200176*((1/a)^n)^5*a^3-5398120*((1/a)^n)^5*a^2-2302804*((1/a)^n)^5*a-8723858* a^20*(1/a)^n+29099952*((1/a)^n)^2*a^18-8422342*a^19*(1/a)^n+32531828*((1/a)^n)^ 2*a^17-6930086*(1/a)^n*a^18-66029712*((1/a)^n)^3*a^15+467236*((1/a)^n)^6+730357 *a^16+40353*a^15-628544*((1/a)^n)^5-7108396*a^14*(1/a)^n+21236684*((1/a)^n)^2*a ^12-6025374*a^13*(1/a)^n+18409160*((1/a)^n)^2*a^11-4109970*(1/a)^n*a^12-\ 34353368*((1/a)^n)^3*a^9+14931000*((1/a)^n)^2*a^10-2480364*(1/a)^n*a^11-\ 26145504*((1/a)^n)^3*a^8+11036760*((1/a)^n)^2*a^9-1753690*(1/a)^n*a^10-19057192 *((1/a)^n)^3*a^7+7366968*((1/a)^n)^2*a^8+18749430*((1/a)^n)^4*a^5-12952688*((1/ a)^n)^3*a^6+11722410*((1/a)^n)^4*a^4-8003240*((1/a)^n)^3*a^5+6610310*((1/a)^n)^ 4*a^3+3269420*((1/a)^n)^4*a^2+1365560*((1/a)^n)^4*a+247661*a^10+502394*a^11+ 149314*a^13+547852*a^12-156579*a^14+1689325*a^18+1491577*a^17-35497*a^20+854088 *a^19+403270*((1/a)^n)^4-123368*((1/a)^n)^3-114736*a^5*(1/a)^n+452228*a^3*((1/a )^n)^2-59948*a^4*(1/a)^n+148820*((1/a)^n)^2*a^2-47680*a^3*(1/a)^n+33656*((1/a)^ n)^2*a+15400*((1/a)^n)^2-13900*a^2*(1/a)^n-286*a*(1/a)^n-1520898*a^9*(1/a)^n+ 4714220*((1/a)^n)^2*a^7-1297422*a^8*(1/a)^n+2884952*((1/a)^n)^2*a^6-845082*(1/a )^n*a^7-4476080*((1/a)^n)^3*a^4+1770440*((1/a)^n)^2*a^5-352842*(1/a)^n*a^6-\ 2314816*((1/a)^n)^3*a^3+960652*((1/a)^n)^2*a^4-976528*((1/a)^n)^3*a^2-364336*(( 1/a)^n)^3*a-500*(1/a)^n-4*a-1803302*a^27*(1/a)^n+4696972*((1/a)^n)^2*a^25-\ 2545260*a^26*(1/a)^n+8597848*((1/a)^n)^2*a^24-2476998*(1/a)^n*a^25-14103208*((1 /a)^n)^3*a^22+12532828*((1/a)^n)^2*a^23-2477178*(1/a)^n*a^24-23084488*((1/a)^n) ^3*a^21+15814988*((1/a)^n)^2*a^22-3363464*(1/a)^n*a^23-33542880*((1/a)^n)^3*a^ 20+18252136*((1/a)^n)^2*a^21-5110352*(1/a)^n*a^22+38291540*((1/a)^n)^4*a^18-\ 43372112*((1/a)^n)^3*a^19+20956180*((1/a)^n)^2*a^20-7277980*(1/a)^n*a^21+ 52970400*((1/a)^n)^4*a^17-51400048*((1/a)^n)^3*a^18+24597832*((1/a)^n)^2*a^19+ 66931060*((1/a)^n)^4*a^16-57512168*((1/a)^n)^3*a^17+78357090*((1/a)^n)^4*a^15-\ 62347936*((1/a)^n)^3*a^16-61529188*((1/a)^n)^5*a^13+85399370*((1/a)^n)^4*a^14-\ 68284776*((1/a)^n)^5*a^12+87897670*((1/a)^n)^4*a^13-70861756*((1/a)^n)^5*a^11+ 86539530*((1/a)^n)^4*a^12-69190296*((1/a)^n)^5*a^10-63630952*((1/a)^n)^5*a^9+ 26858720*((1/a)^n)^6*a^7-54993204*((1/a)^n)^5*a^8+22430324*((1/a)^n)^6*a^6-\ 44650452*((1/a)^n)^5*a^7+17327016*((1/a)^n)^6*a^5+12290936*((1/a)^n)^6*a^4+ 7798140*((1/a)^n)^6*a^3+4238500*((1/a)^n)^6*a^2+1802192*((1/a)^n)^6*a+361427*a^ 22-223669*a^21+506*a^2+281*a^3-1496*a^4+26845*a^7+34497*a^6+14021*a^5+23898*a^9 -34556*a^8-133496*((1/a)^n)^7-43841*a^28+52136*a^26-314366*a^27-3617*a^33+65315 *a^31-13418*a^36+496*a^37+214256*a^30+263958*a^29+1193247*a^24+869546*a^25)^2/( a^2-a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^6+a^3+1)^2/(a^4+1)^2/(a^6+a^5+a^4+a^3+a^2+a+1 )^2/(a^2+1)^4/(a^2+a+1)^6*(a-1)*(a+1))^(1/2) as n goes to infinity, the limiting, 9, scaled moment about the mean is / 24 23 22 21 20 19 18 |- 64 (a - 1) (a + 3 a - 22 a - 180 a - 661 a - 1522 a - 2274 a \ 17 16 15 14 13 12 - 1488 a + 3154 a + 13655 a + 30121 a + 49417 a + 64746 a 11 10 9 8 7 6 + 67169 a + 50393 a + 13291 a - 38902 a - 95064 a - 140286 a 5 4 3 2 2 - 161234 a - 156425 a - 130716 a - 91778 a - 50061 a - 16687) 3 / 2 2 2 2 4 3 2 2 (a + 1) / ((a + 1) (a - a + 1) (a + a + a + a + 1) / 6 3 2 6 5 4 3 2 2 2 6 \1/2 (a + a + 1) (a + a + a + a + a + a + 1) (a + a + 1) )| / and in Maple notation (-64*(a-1)*(a^24+3*a^23-22*a^22-180*a^21-661*a^20-1522*a^19-2274*a^18-1488*a^17 +3154*a^16+13655*a^15+30121*a^14+49417*a^13+64746*a^12+67169*a^11+50393*a^10+ 13291*a^9-38902*a^8-95064*a^7-140286*a^6-161234*a^5-156425*a^4-130716*a^3-91778 *a^2-50061*a-16687)^2*(a+1)^3/(a^2+1)^2/(a^2-a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^6+a^ 3+1)^2/(a^6+a^5+a^4+a^3+a^2+a+1)^2/(a^2+a+1)^6)^(1/2) Here is a plot from a=0 to a=1 *HHHHHH + HHHHHH 120000 HHHH + HHHH + HHH 100000 HHH + HHH + HHH 80000 HHH + HHH + HHH 60000 HHH + HHH + HHH 40000 HHHH + HHH + HHHH 20000 HHHH + HHHHHH + HHHHHHHHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--+--+************ 0 0.2 0.4 0.6 0.8 1 The scaled, 10, th moment about the mean is is n 4 44 n 3 45 n 2 46 n 47 (1 + 126 ((1/a) ) a - 84 ((1/a) ) a + 36 ((1/a) ) a - 9 (1/a) a n 4 43 n 3 44 n 2 45 - 11130 ((1/a) ) a + 13860 ((1/a) ) a - 10890 ((1/a) ) a n 46 n 4 42 n 3 43 + 4965 (1/a) a - 26586 ((1/a) ) a - 53676 ((1/a) ) a n 2 44 n 45 n 5 40 + 139734 ((1/a) ) a - 129111 (1/a) a + 65016 ((1/a) ) a n 4 41 n 3 42 n 2 43 + 174384 ((1/a) ) a - 407316 ((1/a) ) a + 83184 ((1/a) ) a n 5 39 n 4 40 n 3 41 - 232218 ((1/a) ) a + 1056804 ((1/a) ) a - 830676 ((1/a) ) a n 5 38 n 4 39 n 3 40 - 1806588 ((1/a) ) a + 3385032 ((1/a) ) a - 1306452 ((1/a) ) a n 5 37 n 4 38 n 5 36 - 5914566 ((1/a) ) a + 6450990 ((1/a) ) a - 13122774 ((1/a) ) a n 4 37 n 6 34 n 5 35 + 7766304 ((1/a) ) a + 13143060 ((1/a) ) a - 21532644 ((1/a) ) a n 4 36 n 6 33 n 5 34 + 2113524 ((1/a) ) a + 24346392 ((1/a) ) a - 24865470 ((1/a) ) a n 6 32 n 5 33 n 6 31 + 35697816 ((1/a) ) a - 14669172 ((1/a) ) a + 40585356 ((1/a) ) a n 5 32 n 6 30 n 5 31 + 17052336 ((1/a) ) a + 30122400 ((1/a) ) a + 74533410 ((1/a) ) a n 6 29 n 7 27 n 6 28 - 4165560 ((1/a) ) a - 24920208 ((1/a) ) a - 65548224 ((1/a) ) a n 7 26 n 6 27 n 7 25 - 4307868 ((1/a) ) a - 151669812 ((1/a) ) a + 32123088 ((1/a) ) a n 6 26 n 7 24 - 255271716 ((1/a) ) a + 83037564 ((1/a) ) a n 6 25 n 7 23 - 365357160 ((1/a) ) a + 143190684 ((1/a) ) a n 7 22 n 7 21 + 203977944 ((1/a) ) a + 256245732 ((1/a) ) a n 8 19 n 7 20 - 49018347 ((1/a) ) a + 291246948 ((1/a) ) a n 8 18 n 7 19 - 60379677 ((1/a) ) a + 301473576 ((1/a) ) a n 8 17 n 7 18 - 66268332 ((1/a) ) a + 282061836 ((1/a) ) a n 8 16 n 8 15 n 8 14 - 64775817 ((1/a) ) a - 55241901 ((1/a) ) a - 38541456 ((1/a) ) a n 8 13 n 8 12 n 8 11 - 16587396 ((1/a) ) a + 7852536 ((1/a) ) a + 31475295 ((1/a) ) a n 8 10 n 8 25 n 8 32 + 51042204 ((1/a) ) a + 9451611 ((1/a) ) a - 29025 ((1/a) ) a n 7 33 n 8 31 n 7 32 - 2284128 ((1/a) ) a + 295200 ((1/a) ) a - 6256260 ((1/a) ) a n 8 30 n 7 31 n 8 29 + 1078956 ((1/a) ) a - 12757104 ((1/a) ) a + 2494935 ((1/a) ) a n 7 30 n 8 28 n 7 29 - 21073284 ((1/a) ) a + 4545864 ((1/a) ) a - 28783836 ((1/a) ) a n 8 27 n 7 28 n 8 26 + 6913116 ((1/a) ) a - 31696560 ((1/a) ) a + 8891136 ((1/a) ) a n 8 39 n 7 40 n 6 41 + 36 ((1/a) ) a + 180 ((1/a) ) a - 1680 ((1/a) ) a n 5 42 n 8 38 n 7 39 + 5670 ((1/a) ) a - 234 ((1/a) ) a + 4392 ((1/a) ) a n 6 40 n 5 41 n 8 37 - 19824 ((1/a) ) a + 40446 ((1/a) ) a - 2565 ((1/a) ) a n 7 38 n 6 39 n 8 36 + 25956 ((1/a) ) a - 80388 ((1/a) ) a - 11880 ((1/a) ) a n 7 37 n 6 38 n 8 35 + 86580 ((1/a) ) a - 152544 ((1/a) ) a - 35379 ((1/a) ) a n 7 36 n 6 37 n 8 34 + 173916 ((1/a) ) a + 82236 ((1/a) ) a - 73647 ((1/a) ) a n 7 35 n 6 36 n 8 33 + 139824 ((1/a) ) a + 1473444 ((1/a) ) a - 99603 ((1/a) ) a n 7 34 n 6 35 n 8 40 - 432000 ((1/a) ) a + 5396076 ((1/a) ) a + 9 ((1/a) ) a n 7 41 n 6 42 n 5 43 - 36 ((1/a) ) a + 84 ((1/a) ) a - 126 ((1/a) ) a n 8 4 n 8 3 n 8 2 + 35654760 ((1/a) ) a + 23139315 ((1/a) ) a + 12682134 ((1/a) ) a n 8 n 6 20 n 6 19 + 5339844 ((1/a) ) a - 578387124 ((1/a) ) a - 483024696 ((1/a) ) a n 7 17 n 6 18 + 232534476 ((1/a) ) a - 333224556 ((1/a) ) a n 7 16 n 6 17 + 157516992 ((1/a) ) a - 139888980 ((1/a) ) a n 7 15 n 6 16 n 7 14 + 65410308 ((1/a) ) a + 80548776 ((1/a) ) a - 36174672 ((1/a) ) a n 7 13 n 7 12 - 137082960 ((1/a) ) a - 226162044 ((1/a) ) a n 7 11 n 8 9 - 294302196 ((1/a) ) a + 64050480 ((1/a) ) a n 7 10 n 8 8 n 7 9 - 335758896 ((1/a) ) a + 69319575 ((1/a) ) a - 348844500 ((1/a) ) a n 8 7 n 8 6 n 8 5 + 67304853 ((1/a) ) a + 59802057 ((1/a) ) a + 48547269 ((1/a) ) a n 44 n 2 42 n 43 + 504684 (1/a) a - 538146 ((1/a) ) a + 74964 (1/a) a n 2 41 n 42 n 3 39 - 566352 ((1/a) ) a - 827037 (1/a) a + 587244 ((1/a) ) a n 2 40 n 41 n 3 38 - 2568180 ((1/a) ) a + 798420 (1/a) a + 5875548 ((1/a) ) a n 2 39 n 40 n 3 37 - 4554108 ((1/a) ) a + 1204263 (1/a) a + 16889124 ((1/a) ) a n 2 38 n 39 n 4 35 - 7742022 ((1/a) ) a + 629247 (1/a) a - 16320276 ((1/a) ) a n 3 36 n 2 37 n 38 + 30402624 ((1/a) ) a - 7524306 ((1/a) ) a + 2518413 (1/a) a n 4 34 n 3 35 n 2 36 - 48106506 ((1/a) ) a + 44281608 ((1/a) ) a - 14813772 ((1/a) ) a n 37 n 4 33 n 3 34 + 7346613 (1/a) a - 92173578 ((1/a) ) a + 64120644 ((1/a) ) a n 2 35 n 36 n 4 32 - 29703792 ((1/a) ) a + 6830283 (1/a) a - 145062540 ((1/a) ) a n 3 33 n 2 34 + 93621108 ((1/a) ) a - 49094970 ((1/a) ) a n 5 30 n 4 31 + 156335886 ((1/a) ) a - 206718666 ((1/a) ) a n 3 32 n 5 29 + 137943624 ((1/a) ) a + 253410066 ((1/a) ) a n 4 30 n 3 31 - 278655762 ((1/a) ) a + 192474828 ((1/a) ) a n 5 28 n 4 29 + 360125262 ((1/a) ) a - 367995852 ((1/a) ) a n 5 27 n 4 28 + 471723840 ((1/a) ) a - 464574684 ((1/a) ) a n 5 26 n 4 27 + 582734376 ((1/a) ) a - 551618592 ((1/a) ) a n 6 24 n 5 25 - 468297564 ((1/a) ) a + 681190902 ((1/a) ) a n 6 23 n 5 24 - 550414284 ((1/a) ) a + 750712158 ((1/a) ) a n 6 22 n 5 23 - 603385776 ((1/a) ) a + 772984296 ((1/a) ) a n 6 21 n 5 22 n 8 - 616397460 ((1/a) ) a + 721413882 ((1/a) ) a + 1334961 ((1/a) ) n 7 7 n 6 8 n 7 6 - 303011280 ((1/a) ) a + 692114220 ((1/a) ) a - 253918944 ((1/a) ) a n 7 5 n 7 4 n 7 3 - 197628516 ((1/a) ) a - 140725260 ((1/a) ) a - 89553996 ((1/a) ) a n 7 2 n 7 35 n - 48726072 ((1/a) ) a - 20691900 ((1/a) ) a - 1932831 a (1/a) n 2 33 34 n n 2 32 - 56095878 ((1/a) ) a - 7490136 a (1/a) - 50842482 ((1/a) ) a n 33 n 3 30 n 2 31 - 1609479 (1/a) a + 240072252 ((1/a) ) a - 38160828 ((1/a) ) a n 32 n 3 29 n 2 30 + 8540337 (1/a) a + 260622096 ((1/a) ) a - 30049380 ((1/a) ) a n 31 n 3 28 n 2 29 + 13948095 (1/a) a + 248550624 ((1/a) ) a - 30833166 ((1/a) ) a n 30 n 4 26 n 3 27 + 10918293 (1/a) a - 598979178 ((1/a) ) a + 208452216 ((1/a) ) a n 2 28 n 29 n 4 25 - 38896578 ((1/a) ) a - 6779388 (1/a) a - 586358304 ((1/a) ) a n 3 26 n 2 27 n 28 + 158974200 ((1/a) ) a - 38295186 ((1/a) ) a - 35941950 (1/a) a n 4 24 n 3 25 - 501305238 ((1/a) ) a + 109064676 ((1/a) ) a n 2 26 n 4 23 n 3 24 - 4735890 ((1/a) ) a - 357196854 ((1/a) ) a + 49775544 ((1/a) ) a n 5 21 n 4 22 + 585342198 ((1/a) ) a - 177016560 ((1/a) ) a n 3 23 n 5 20 - 35724612 ((1/a) ) a + 372390984 ((1/a) ) a n 4 21 n 5 19 + 18955734 ((1/a) ) a + 104701842 ((1/a) ) a n 4 20 n 5 18 + 231111678 ((1/a) ) a - 193218102 ((1/a) ) a n 4 19 n 5 17 + 456561504 ((1/a) ) a - 491520456 ((1/a) ) a n 6 15 n 5 16 + 311021592 ((1/a) ) a - 757569960 ((1/a) ) a n 6 14 n 5 15 + 522676224 ((1/a) ) a - 972759942 ((1/a) ) a n 6 13 n 5 14 + 693354060 ((1/a) ) a - 1133860266 ((1/a) ) a n 6 12 n 6 11 + 810127920 ((1/a) ) a + 865368504 ((1/a) ) a n 6 10 n 7 8 + 857348604 ((1/a) ) a - 336121632 ((1/a) ) a n 6 9 32 35 23 + 794194548 ((1/a) ) a + 9996828 a - 7236981 a + 16830586 a 40 38 47 42 34 - 689152 a + 2541257 a - 1007 a - 127880 a - 6255786 a n 8 24 n 8 23 n 8 22 + 7446447 ((1/a) ) a + 1930212 ((1/a) ) a - 7455573 ((1/a) ) a n 8 21 n 8 20 - 20163843 ((1/a) ) a - 34745130 ((1/a) ) a n 2 16 n 17 n 3 14 + 299778030 ((1/a) ) a - 39026973 (1/a) a - 673471428 ((1/a) ) a n 2 15 n 16 n 3 13 + 265459008 ((1/a) ) a - 33847875 (1/a) a - 621340524 ((1/a) ) a n 2 14 n 15 n 4 11 + 217841292 ((1/a) ) a - 38553477 (1/a) a + 949546458 ((1/a) ) a n 3 12 n 2 13 - 544299168 ((1/a) ) a + 172238238 ((1/a) ) a n 4 10 n 3 11 + 801793986 ((1/a) ) a - 448221144 ((1/a) ) a n 4 9 n 3 10 + 650138916 ((1/a) ) a - 349557768 ((1/a) ) a n 4 8 n 5 6 n 4 7 + 503078436 ((1/a) ) a - 435523914 ((1/a) ) a + 367850196 ((1/a) ) a n 5 5 n 4 6 n 5 4 - 301236012 ((1/a) ) a + 248692710 ((1/a) ) a - 192099222 ((1/a) ) a n 5 3 n 5 2 n 5 - 111600216 ((1/a) ) a - 56679966 ((1/a) ) a - 23890230 ((1/a) ) a 20 n n 2 18 19 n - 100984944 a (1/a) + 283995198 ((1/a) ) a - 89652150 a (1/a) n 2 17 n 18 n 3 15 + 304315746 ((1/a) ) a - 61743792 (1/a) a - 704513292 ((1/a) ) a n 6 16 15 n 5 + 8510376 ((1/a) ) + 8206800 a + 1260662 a - 6841674 ((1/a) ) 14 n n 2 12 13 n - 42831441 a (1/a) + 137771250 ((1/a) ) a - 38954124 a (1/a) n 2 11 n 12 n 3 9 + 111887382 ((1/a) ) a - 26593209 (1/a) a - 259173264 ((1/a) ) a n 2 10 n 11 n 3 8 + 88066572 ((1/a) ) a - 13349583 (1/a) a - 184488024 ((1/a) ) a n 2 9 n 10 n 3 7 + 64368336 ((1/a) ) a - 6447033 (1/a) a - 127365168 ((1/a) ) a n 2 8 n 4 5 n 3 6 + 41607402 ((1/a) ) a + 157796688 ((1/a) ) a - 82487244 ((1/a) ) a n 4 4 n 3 5 n 4 3 + 93940896 ((1/a) ) a - 48945708 ((1/a) ) a + 50293236 ((1/a) ) a n 4 2 n 4 10 11 + 23045526 ((1/a) ) a + 9162090 ((1/a) ) a + 1369535 a + 2218156 a 13 12 14 18 17 - 1444089 a + 1555575 a - 2357286 a + 14360290 a + 14324295 a 20 19 n 4 n 3 - 6551946 a + 3828799 a + 2885694 ((1/a) ) - 598416 ((1/a) ) 5 n 3 n 2 4 n - 419703 a (1/a) + 1998306 a ((1/a) ) - 117864 a (1/a) n 2 2 3 n n 2 + 621036 ((1/a) ) a - 139044 a (1/a) + 90570 ((1/a) ) a n 2 2 n n 9 n + 49914 ((1/a) ) - 46089 a (1/a) - 45 a (1/a) - 4733613 a (1/a) n 2 7 8 n n 2 6 + 25058178 ((1/a) ) a - 4787967 a (1/a) + 13541100 ((1/a) ) a n 7 n 3 4 n 2 5 - 3749643 (1/a) a - 26326524 ((1/a) ) a + 7584192 ((1/a) ) a n 6 n 3 3 n 2 4 - 1648860 (1/a) a - 13245624 ((1/a) ) a + 3983196 ((1/a) ) a n 3 2 n 3 n - 5014884 ((1/a) ) a - 1604400 ((1/a) ) a - 1011 (1/a) - 5 a 27 n n 2 25 26 n - 58864356 a (1/a) + 63014778 ((1/a) ) a - 57068913 a (1/a) n 2 24 n 25 n 3 22 + 143689116 ((1/a) ) a - 36131310 (1/a) a - 164530128 ((1/a) ) a n 2 23 n 24 n 3 21 + 202924950 ((1/a) ) a - 21570747 (1/a) a - 327125904 ((1/a) ) a n 2 22 n 23 n 3 20 + 230562774 ((1/a) ) a - 30297213 (1/a) a - 488722416 ((1/a) ) a n 2 21 n 22 n 4 18 + 233673132 ((1/a) ) a - 57224370 (1/a) a + 684207258 ((1/a) ) a n 3 19 n 2 20 n 21 - 616954380 ((1/a) ) a + 237689970 ((1/a) ) a - 87127287 (1/a) a n 4 17 n 3 18 + 899947692 ((1/a) ) a - 693855876 ((1/a) ) a n 2 19 n 4 16 + 253557306 ((1/a) ) a + 1077334944 ((1/a) ) a n 3 17 n 4 15 - 721458444 ((1/a) ) a + 1186971492 ((1/a) ) a n 3 16 n 5 13 - 721227276 ((1/a) ) a - 1232602686 ((1/a) ) a n 4 14 n 5 12 + 1213663122 ((1/a) ) a - 1258535250 ((1/a) ) a n 4 13 n 5 11 + 1170208326 ((1/a) ) a - 1212333696 ((1/a) ) a n 4 12 n 5 10 + 1075396560 ((1/a) ) a - 1107257508 ((1/a) ) a n 5 9 n 6 7 n 5 8 - 957992490 ((1/a) ) a + 574014924 ((1/a) ) a - 780018876 ((1/a) ) a n 6 6 n 5 7 n 6 5 + 452213496 ((1/a) ) a - 598206546 ((1/a) ) a + 332493924 ((1/a) ) a n 6 4 n 6 3 n 6 2 + 226247784 ((1/a) ) a + 139057128 ((1/a) ) a + 74118324 ((1/a) ) a n 6 22 21 2 3 + 31594080 ((1/a) ) a + 2059984 a - 7755557 a + 1021 a + 34 a 4 7 6 5 9 8 - 3816 a + 25797 a + 95248 a + 48467 a + 248993 a - 205150 a n 7 28 26 27 - 5339844 ((1/a) ) - 12210294 a + 5536570 a - 10122833 a 41 39 45 33 31 - 112395 a - 338911 a - 413816 a + 2999348 a + 13596873 a 36 37 30 48 29 - 834435 a + 4156276 a + 9777058 a + a - 1017281 a 24 25 46 44 43 / + 26226822 a + 23288164 a + 45889 a + 1101066 a - 858775 a ) / / n 4 n 4 2 3 2 2 2 (((1/a) - 1) ((1/a) - a) (a + a + 1) (a + 1) (a - a + 1) 4 3 2 2 4 6 5 4 3 2 (a + a + a + a + 1) (a + 1) (a + a + a + a + a + a + 1) 6 3 4 3 2 (a + a + 1) (a - a + a - a + 1)) and in Maple notation 1/((1/a)^n-1)^4/((1/a)^n-a)^4*(1+126*((1/a)^n)^4*a^44-84*((1/a)^n)^3*a^45+36*(( 1/a)^n)^2*a^46-9*(1/a)^n*a^47-11130*((1/a)^n)^4*a^43+13860*((1/a)^n)^3*a^44-\ 10890*((1/a)^n)^2*a^45+4965*(1/a)^n*a^46-26586*((1/a)^n)^4*a^42-53676*((1/a)^n) ^3*a^43+139734*((1/a)^n)^2*a^44-129111*(1/a)^n*a^45+65016*((1/a)^n)^5*a^40+ 174384*((1/a)^n)^4*a^41-407316*((1/a)^n)^3*a^42+83184*((1/a)^n)^2*a^43-232218*( (1/a)^n)^5*a^39+1056804*((1/a)^n)^4*a^40-830676*((1/a)^n)^3*a^41-1806588*((1/a) ^n)^5*a^38+3385032*((1/a)^n)^4*a^39-1306452*((1/a)^n)^3*a^40-5914566*((1/a)^n)^ 5*a^37+6450990*((1/a)^n)^4*a^38-13122774*((1/a)^n)^5*a^36+7766304*((1/a)^n)^4*a ^37+13143060*((1/a)^n)^6*a^34-21532644*((1/a)^n)^5*a^35+2113524*((1/a)^n)^4*a^ 36+24346392*((1/a)^n)^6*a^33-24865470*((1/a)^n)^5*a^34+35697816*((1/a)^n)^6*a^ 32-14669172*((1/a)^n)^5*a^33+40585356*((1/a)^n)^6*a^31+17052336*((1/a)^n)^5*a^ 32+30122400*((1/a)^n)^6*a^30+74533410*((1/a)^n)^5*a^31-4165560*((1/a)^n)^6*a^29 -24920208*((1/a)^n)^7*a^27-65548224*((1/a)^n)^6*a^28-4307868*((1/a)^n)^7*a^26-\ 151669812*((1/a)^n)^6*a^27+32123088*((1/a)^n)^7*a^25-255271716*((1/a)^n)^6*a^26 +83037564*((1/a)^n)^7*a^24-365357160*((1/a)^n)^6*a^25+143190684*((1/a)^n)^7*a^ 23+203977944*((1/a)^n)^7*a^22+256245732*((1/a)^n)^7*a^21-49018347*((1/a)^n)^8*a ^19+291246948*((1/a)^n)^7*a^20-60379677*((1/a)^n)^8*a^18+301473576*((1/a)^n)^7* a^19-66268332*((1/a)^n)^8*a^17+282061836*((1/a)^n)^7*a^18-64775817*((1/a)^n)^8* a^16-55241901*((1/a)^n)^8*a^15-38541456*((1/a)^n)^8*a^14-16587396*((1/a)^n)^8*a ^13+7852536*((1/a)^n)^8*a^12+31475295*((1/a)^n)^8*a^11+51042204*((1/a)^n)^8*a^ 10+9451611*((1/a)^n)^8*a^25-29025*((1/a)^n)^8*a^32-2284128*((1/a)^n)^7*a^33+ 295200*((1/a)^n)^8*a^31-6256260*((1/a)^n)^7*a^32+1078956*((1/a)^n)^8*a^30-\ 12757104*((1/a)^n)^7*a^31+2494935*((1/a)^n)^8*a^29-21073284*((1/a)^n)^7*a^30+ 4545864*((1/a)^n)^8*a^28-28783836*((1/a)^n)^7*a^29+6913116*((1/a)^n)^8*a^27-\ 31696560*((1/a)^n)^7*a^28+8891136*((1/a)^n)^8*a^26+36*((1/a)^n)^8*a^39+180*((1/ a)^n)^7*a^40-1680*((1/a)^n)^6*a^41+5670*((1/a)^n)^5*a^42-234*((1/a)^n)^8*a^38+ 4392*((1/a)^n)^7*a^39-19824*((1/a)^n)^6*a^40+40446*((1/a)^n)^5*a^41-2565*((1/a) ^n)^8*a^37+25956*((1/a)^n)^7*a^38-80388*((1/a)^n)^6*a^39-11880*((1/a)^n)^8*a^36 +86580*((1/a)^n)^7*a^37-152544*((1/a)^n)^6*a^38-35379*((1/a)^n)^8*a^35+173916*( (1/a)^n)^7*a^36+82236*((1/a)^n)^6*a^37-73647*((1/a)^n)^8*a^34+139824*((1/a)^n)^ 7*a^35+1473444*((1/a)^n)^6*a^36-99603*((1/a)^n)^8*a^33-432000*((1/a)^n)^7*a^34+ 5396076*((1/a)^n)^6*a^35+9*((1/a)^n)^8*a^40-36*((1/a)^n)^7*a^41+84*((1/a)^n)^6* a^42-126*((1/a)^n)^5*a^43+35654760*((1/a)^n)^8*a^4+23139315*((1/a)^n)^8*a^3+ 12682134*((1/a)^n)^8*a^2+5339844*((1/a)^n)^8*a-578387124*((1/a)^n)^6*a^20-\ 483024696*((1/a)^n)^6*a^19+232534476*((1/a)^n)^7*a^17-333224556*((1/a)^n)^6*a^ 18+157516992*((1/a)^n)^7*a^16-139888980*((1/a)^n)^6*a^17+65410308*((1/a)^n)^7*a ^15+80548776*((1/a)^n)^6*a^16-36174672*((1/a)^n)^7*a^14-137082960*((1/a)^n)^7*a ^13-226162044*((1/a)^n)^7*a^12-294302196*((1/a)^n)^7*a^11+64050480*((1/a)^n)^8* a^9-335758896*((1/a)^n)^7*a^10+69319575*((1/a)^n)^8*a^8-348844500*((1/a)^n)^7*a ^9+67304853*((1/a)^n)^8*a^7+59802057*((1/a)^n)^8*a^6+48547269*((1/a)^n)^8*a^5+ 504684*(1/a)^n*a^44-538146*((1/a)^n)^2*a^42+74964*(1/a)^n*a^43-566352*((1/a)^n) ^2*a^41-827037*(1/a)^n*a^42+587244*((1/a)^n)^3*a^39-2568180*((1/a)^n)^2*a^40+ 798420*(1/a)^n*a^41+5875548*((1/a)^n)^3*a^38-4554108*((1/a)^n)^2*a^39+1204263*( 1/a)^n*a^40+16889124*((1/a)^n)^3*a^37-7742022*((1/a)^n)^2*a^38+629247*(1/a)^n*a ^39-16320276*((1/a)^n)^4*a^35+30402624*((1/a)^n)^3*a^36-7524306*((1/a)^n)^2*a^ 37+2518413*(1/a)^n*a^38-48106506*((1/a)^n)^4*a^34+44281608*((1/a)^n)^3*a^35-\ 14813772*((1/a)^n)^2*a^36+7346613*(1/a)^n*a^37-92173578*((1/a)^n)^4*a^33+ 64120644*((1/a)^n)^3*a^34-29703792*((1/a)^n)^2*a^35+6830283*(1/a)^n*a^36-\ 145062540*((1/a)^n)^4*a^32+93621108*((1/a)^n)^3*a^33-49094970*((1/a)^n)^2*a^34+ 156335886*((1/a)^n)^5*a^30-206718666*((1/a)^n)^4*a^31+137943624*((1/a)^n)^3*a^ 32+253410066*((1/a)^n)^5*a^29-278655762*((1/a)^n)^4*a^30+192474828*((1/a)^n)^3* a^31+360125262*((1/a)^n)^5*a^28-367995852*((1/a)^n)^4*a^29+471723840*((1/a)^n)^ 5*a^27-464574684*((1/a)^n)^4*a^28+582734376*((1/a)^n)^5*a^26-551618592*((1/a)^n )^4*a^27-468297564*((1/a)^n)^6*a^24+681190902*((1/a)^n)^5*a^25-550414284*((1/a) ^n)^6*a^23+750712158*((1/a)^n)^5*a^24-603385776*((1/a)^n)^6*a^22+772984296*((1/ a)^n)^5*a^23-616397460*((1/a)^n)^6*a^21+721413882*((1/a)^n)^5*a^22+1334961*((1/ a)^n)^8-303011280*((1/a)^n)^7*a^7+692114220*((1/a)^n)^6*a^8-253918944*((1/a)^n) ^7*a^6-197628516*((1/a)^n)^7*a^5-140725260*((1/a)^n)^7*a^4-89553996*((1/a)^n)^7 *a^3-48726072*((1/a)^n)^7*a^2-20691900*((1/a)^n)^7*a-1932831*a^35*(1/a)^n-\ 56095878*((1/a)^n)^2*a^33-7490136*a^34*(1/a)^n-50842482*((1/a)^n)^2*a^32-\ 1609479*(1/a)^n*a^33+240072252*((1/a)^n)^3*a^30-38160828*((1/a)^n)^2*a^31+ 8540337*(1/a)^n*a^32+260622096*((1/a)^n)^3*a^29-30049380*((1/a)^n)^2*a^30+ 13948095*(1/a)^n*a^31+248550624*((1/a)^n)^3*a^28-30833166*((1/a)^n)^2*a^29+ 10918293*(1/a)^n*a^30-598979178*((1/a)^n)^4*a^26+208452216*((1/a)^n)^3*a^27-\ 38896578*((1/a)^n)^2*a^28-6779388*(1/a)^n*a^29-586358304*((1/a)^n)^4*a^25+ 158974200*((1/a)^n)^3*a^26-38295186*((1/a)^n)^2*a^27-35941950*(1/a)^n*a^28-\ 501305238*((1/a)^n)^4*a^24+109064676*((1/a)^n)^3*a^25-4735890*((1/a)^n)^2*a^26-\ 357196854*((1/a)^n)^4*a^23+49775544*((1/a)^n)^3*a^24+585342198*((1/a)^n)^5*a^21 -177016560*((1/a)^n)^4*a^22-35724612*((1/a)^n)^3*a^23+372390984*((1/a)^n)^5*a^ 20+18955734*((1/a)^n)^4*a^21+104701842*((1/a)^n)^5*a^19+231111678*((1/a)^n)^4*a ^20-193218102*((1/a)^n)^5*a^18+456561504*((1/a)^n)^4*a^19-491520456*((1/a)^n)^5 *a^17+311021592*((1/a)^n)^6*a^15-757569960*((1/a)^n)^5*a^16+522676224*((1/a)^n) ^6*a^14-972759942*((1/a)^n)^5*a^15+693354060*((1/a)^n)^6*a^13-1133860266*((1/a) ^n)^5*a^14+810127920*((1/a)^n)^6*a^12+865368504*((1/a)^n)^6*a^11+857348604*((1/ a)^n)^6*a^10-336121632*((1/a)^n)^7*a^8+794194548*((1/a)^n)^6*a^9+9996828*a^32-\ 7236981*a^35+16830586*a^23-689152*a^40+2541257*a^38-1007*a^47-127880*a^42-\ 6255786*a^34+7446447*((1/a)^n)^8*a^24+1930212*((1/a)^n)^8*a^23-7455573*((1/a)^n )^8*a^22-20163843*((1/a)^n)^8*a^21-34745130*((1/a)^n)^8*a^20+299778030*((1/a)^n )^2*a^16-39026973*(1/a)^n*a^17-673471428*((1/a)^n)^3*a^14+265459008*((1/a)^n)^2 *a^15-33847875*(1/a)^n*a^16-621340524*((1/a)^n)^3*a^13+217841292*((1/a)^n)^2*a^ 14-38553477*(1/a)^n*a^15+949546458*((1/a)^n)^4*a^11-544299168*((1/a)^n)^3*a^12+ 172238238*((1/a)^n)^2*a^13+801793986*((1/a)^n)^4*a^10-448221144*((1/a)^n)^3*a^ 11+650138916*((1/a)^n)^4*a^9-349557768*((1/a)^n)^3*a^10+503078436*((1/a)^n)^4*a ^8-435523914*((1/a)^n)^5*a^6+367850196*((1/a)^n)^4*a^7-301236012*((1/a)^n)^5*a^ 5+248692710*((1/a)^n)^4*a^6-192099222*((1/a)^n)^5*a^4-111600216*((1/a)^n)^5*a^3 -56679966*((1/a)^n)^5*a^2-23890230*((1/a)^n)^5*a-100984944*a^20*(1/a)^n+ 283995198*((1/a)^n)^2*a^18-89652150*a^19*(1/a)^n+304315746*((1/a)^n)^2*a^17-\ 61743792*(1/a)^n*a^18-704513292*((1/a)^n)^3*a^15+8510376*((1/a)^n)^6+8206800*a^ 16+1260662*a^15-6841674*((1/a)^n)^5-42831441*a^14*(1/a)^n+137771250*((1/a)^n)^2 *a^12-38954124*a^13*(1/a)^n+111887382*((1/a)^n)^2*a^11-26593209*(1/a)^n*a^12-\ 259173264*((1/a)^n)^3*a^9+88066572*((1/a)^n)^2*a^10-13349583*(1/a)^n*a^11-\ 184488024*((1/a)^n)^3*a^8+64368336*((1/a)^n)^2*a^9-6447033*(1/a)^n*a^10-\ 127365168*((1/a)^n)^3*a^7+41607402*((1/a)^n)^2*a^8+157796688*((1/a)^n)^4*a^5-\ 82487244*((1/a)^n)^3*a^6+93940896*((1/a)^n)^4*a^4-48945708*((1/a)^n)^3*a^5+ 50293236*((1/a)^n)^4*a^3+23045526*((1/a)^n)^4*a^2+9162090*((1/a)^n)^4*a+1369535 *a^10+2218156*a^11-1444089*a^13+1555575*a^12-2357286*a^14+14360290*a^18+ 14324295*a^17-6551946*a^20+3828799*a^19+2885694*((1/a)^n)^4-598416*((1/a)^n)^3-\ 419703*a^5*(1/a)^n+1998306*a^3*((1/a)^n)^2-117864*a^4*(1/a)^n+621036*((1/a)^n)^ 2*a^2-139044*a^3*(1/a)^n+90570*((1/a)^n)^2*a+49914*((1/a)^n)^2-46089*a^2*(1/a)^ n-45*a*(1/a)^n-4733613*a^9*(1/a)^n+25058178*((1/a)^n)^2*a^7-4787967*a^8*(1/a)^n +13541100*((1/a)^n)^2*a^6-3749643*(1/a)^n*a^7-26326524*((1/a)^n)^3*a^4+7584192* ((1/a)^n)^2*a^5-1648860*(1/a)^n*a^6-13245624*((1/a)^n)^3*a^3+3983196*((1/a)^n)^ 2*a^4-5014884*((1/a)^n)^3*a^2-1604400*((1/a)^n)^3*a-1011*(1/a)^n-5*a-58864356*a ^27*(1/a)^n+63014778*((1/a)^n)^2*a^25-57068913*a^26*(1/a)^n+143689116*((1/a)^n) ^2*a^24-36131310*(1/a)^n*a^25-164530128*((1/a)^n)^3*a^22+202924950*((1/a)^n)^2* a^23-21570747*(1/a)^n*a^24-327125904*((1/a)^n)^3*a^21+230562774*((1/a)^n)^2*a^ 22-30297213*(1/a)^n*a^23-488722416*((1/a)^n)^3*a^20+233673132*((1/a)^n)^2*a^21-\ 57224370*(1/a)^n*a^22+684207258*((1/a)^n)^4*a^18-616954380*((1/a)^n)^3*a^19+ 237689970*((1/a)^n)^2*a^20-87127287*(1/a)^n*a^21+899947692*((1/a)^n)^4*a^17-\ 693855876*((1/a)^n)^3*a^18+253557306*((1/a)^n)^2*a^19+1077334944*((1/a)^n)^4*a^ 16-721458444*((1/a)^n)^3*a^17+1186971492*((1/a)^n)^4*a^15-721227276*((1/a)^n)^3 *a^16-1232602686*((1/a)^n)^5*a^13+1213663122*((1/a)^n)^4*a^14-1258535250*((1/a) ^n)^5*a^12+1170208326*((1/a)^n)^4*a^13-1212333696*((1/a)^n)^5*a^11+1075396560*( (1/a)^n)^4*a^12-1107257508*((1/a)^n)^5*a^10-957992490*((1/a)^n)^5*a^9+574014924 *((1/a)^n)^6*a^7-780018876*((1/a)^n)^5*a^8+452213496*((1/a)^n)^6*a^6-598206546* ((1/a)^n)^5*a^7+332493924*((1/a)^n)^6*a^5+226247784*((1/a)^n)^6*a^4+139057128*( (1/a)^n)^6*a^3+74118324*((1/a)^n)^6*a^2+31594080*((1/a)^n)^6*a+2059984*a^22-\ 7755557*a^21+1021*a^2+34*a^3-3816*a^4+25797*a^7+95248*a^6+48467*a^5+248993*a^9-\ 205150*a^8-5339844*((1/a)^n)^7-12210294*a^28+5536570*a^26-10122833*a^27-112395* a^41-338911*a^39-413816*a^45+2999348*a^33+13596873*a^31-834435*a^36+4156276*a^ 37+9777058*a^30+a^48-1017281*a^29+26226822*a^24+23288164*a^25+45889*a^46+ 1101066*a^44-858775*a^43)/(a^2+a+1)^3/(a^2+1)^2/(a^2-a+1)/(a^4+a^3+a^2+a+1)^2/( a^4+1)/(a^6+a^5+a^4+a^3+a^2+a+1)/(a^6+a^3+1)/(a^4-a^3+a^2-a+1) as n goes to infinity, the limiting, 10, -th scaled moment about the mean is is 32 31 30 29 28 27 26 25 9 (a + 3 a - 30 a - 259 a - 1035 a - 2611 a - 4252 a - 2884 a 24 23 22 21 20 19 + 7842 a + 36026 a + 87087 a + 157301 a + 227622 a + 261993 a 18 17 16 15 14 + 217169 a + 58023 a - 225680 a - 605073 a - 1006839 a 13 12 11 10 9 - 1324943 a - 1462842 a - 1358291 a - 1000377 a - 437126 a 8 7 6 5 4 + 223858 a + 833644 a + 1250532 a + 1432501 a + 1390605 a 3 2 / 2 + 1161909 a + 815810 a + 444987 a + 148329) / ((a - a + 1) / 4 3 2 4 6 5 4 3 2 (a - a + a - a + 1) (a + 1) (a + a + a + a + a + a + 1) 2 2 4 3 2 2 2 2 (a + a + 1) (a + a + a + a + 1) (a + 1) ) and in Maple notation 9*(a^32+3*a^31-30*a^30-259*a^29-1035*a^28-2611*a^27-4252*a^26-2884*a^25+7842*a^ 24+36026*a^23+87087*a^22+157301*a^21+227622*a^20+261993*a^19+217169*a^18+58023* a^17-225680*a^16-605073*a^15-1006839*a^14-1324943*a^13-1462842*a^12-1358291*a^ 11-1000377*a^10-437126*a^9+223858*a^8+833644*a^7+1250532*a^6+1432501*a^5+ 1390605*a^4+1161909*a^3+815810*a^2+444987*a+148329)/(a^2-a+1)/(a^4-a^3+a^2-a+1) /(a^4+1)/(a^6+a^5+a^4+a^3+a^2+a+1)/(a^2+a+1)^2/(a^4+a^3+a^2+a+1)^2/(a^2+1)^2 Here is a plot from a=0 to a=1 *HHHHHH + HHHHH 1.2e+06 HHHH + HHHH + HHH 1e+06 HHH + HHH + HH 800000 HHH + HHH + HHH 600000 HH + HHHH + HHH 400000 HHH + HHHH + HHHH 200000 HHHH + HHHHH + HHHHHHHHH +--+--+--+--+--+--+--+--+--+--+--+--+---+--+--+--+--+--+--+--**************** 0 0.2 0.4 0.6 0.8 1 -------------------------------- This ends this article, that took, 31813.444, to generate.