The Statistical Distribution of the Evil Location of Evil Real Numbers for \ any Base By Shalosh B. Ekhad Definition (Isaac Hodes): A real number is K-evil in base b if its base-b r\ epresentation has the property that one of the partial sums is K Theorem 1: Let a[b](n) be the probability that a random real number in base \ b has one of its partial sums equal to n, then infinity ----- (b - 1) 2 2 \ n (-1 + x ) x ) a[b](n) x = - ------------------------------------ / b ----- (-b x + x + b - 1) (b - 1) (-1 + x) n = 0 and in Maple notation Sum(a[b](n)*x^n,n = 0 .. infinity) = -(-1+x^(b-1))^2*x^2/(-b*x+x^b+b-1)/(b-1)/( -1+x) it follows that as n goes to infinity, the prob. that one of the partial sum\ s of a random real number equals n, goes to 2/b Theorem 2: Let A[b](n,k) be the probability that a random real number in bas\ e b has its k-th partial sum equal to n , then infinity /infinity \ ----- | ----- | (b - 1) 2 2 2 \ | \ n k| (-1 + x ) x t ) | ) A[b](n, k) x t | = - ------------------------------------- / | / | b ----- | ----- | (t x - b x + b - t) (b - 1) (-1 + x) n = 0 \ k = 0 / and in Maple notation Sum(Sum(A[b](n,k)*x^n*t^k,k = 0 .. infinity),n = 0 .. infinity) = -(-1+x^(b-1)) ^2*x^2*t^2/(t*x^b-b*x+b-t)/(b-1)/(-1+x) it follows that as n goes to infinity, the expected evil location of an n-ev\ -5 + b 2 n il number is asymptotic to, --------- + ----- 3 (b - 1) b - 1 and in Maple notation 1/3*(-5+b)/(b-1)+2/(b-1)*n In fact we have explicit expressions for all central moments up to the , 16 The variance is asymptotic (with exponentially decaying error) to, 2 2 (b - 7 b + 1) 2 (1 + b) n ---------------- + ----------- 2 2 9 (b - 1) 3 (b - 1) and in Maple notation 2/9*(b^2-7*b+1)/(b-1)^2+2/3*(1+b)/(b-1)^2*n 2 2 2 (1 + b) (11 b - 95 b + 11) 2 (1 + b) n The skewness is, ----------------------------- + ------------ 3 3 135 (b - 1) 3 (b - 1) and in Maple notation 2/135*(1+b)*(11*b^2-95*b+11)/(b-1)^3+2/3*(1+b)^2/(b-1)^3*n 2 2 2 (17 b - 11 b + 17) (b - 10 b + 1) The kurtosis is, ------------------------------------- 4 135 (b - 1) 2 2 2 2 (1 + b) (13 b - 30 b + 13) n 4 (1 + b) n + ------------------------------- + ------------- 4 4 15 (b - 1) 3 (b - 1) and in Maple notation 2/135*(17*b^2-11*b+17)*(b^2-10*b+1)/(b-1)^4+2/15*(1+b)*(13*b^2-30*b+13)/(b-1)^4 *n+4/3*(1+b)^2/(b-1)^4*n^2 The , 5, -th central moment of an n-evil location for random evil real number\ in base b is 4 3 2 2 (1 + b) (293 b - 4291 b + 7356 b - 4291 b + 293) ----------------------------------------------------- 5 1701 (b - 1) 2 2 3 2 10 (31 b - 118 b + 31) (1 + b) n 40 (1 + b) n + ---------------------------------- + -------------- 5 5 81 (b - 1) 9 (b - 1) and in Maple notation 2/1701*(1+b)*(293*b^4-4291*b^3+7356*b^2-4291*b+293)/(b-1)^5+10/81*(31*b^2-118*b +31)*(1+b)^2/(b-1)^5*n+40/9*(1+b)^3/(b-1)^5*n^2 The , 6, -th central moment of an n-evil location for random evil real number\ in base b is 6 5 4 3 2 2 (5413 b - 120459 b + 116799 b + 77102 b + 116799 b - 120459 b + 5413) / 6 / (25515 (b - 1) ) / 4 3 2 2 (1 + b) (2713 b - 9884 b - 6618 b - 9884 b + 2713) n + --------------------------------------------------------- 6 567 (b - 1) 2 2 2 3 3 52 (b + 1) (1 + b) n 40 (1 + b) n + ----------------------- + -------------- 6 6 3 (b - 1) 9 (b - 1) and in Maple notation 2/25515*(5413*b^6-120459*b^5+116799*b^4+77102*b^3+116799*b^2-120459*b+5413)/(b-\ 1)^6+2/567*(1+b)*(2713*b^4-9884*b^3-6618*b^2-9884*b+2713)/(b-1)^6*n+52/3*(b^2+1 )*(1+b)^2/(b-1)^6*n^2+40/9*(1+b)^3/(b-1)^6*n^3 The , 7, -th central moment of an n-evil location for random evil real number\ in base b is 6 5 4 3 2 2 (1 + b) (125 b - 35517 b + 61143 b - 12494 b + 61143 b - 35517 b + 125) / 7 / (3645 (b - 1) ) / 4 3 2 2 14 (717 b - 3680 b + 854 b - 3680 b + 717) (1 + b) n + -------------------------------------------------------- 7 405 (b - 1) 2 3 2 4 3 280 (19 b - 25 b + 19) (1 + b) n 280 (1 + b) n + ----------------------------------- + --------------- 7 7 81 (b - 1) 9 (b - 1) and in Maple notation 2/3645*(1+b)*(125*b^6-35517*b^5+61143*b^4-12494*b^3+61143*b^2-35517*b+125)/(b-1 )^7+14/405*(717*b^4-3680*b^3+854*b^2-3680*b+717)*(1+b)^2/(b-1)^7*n+280/81*(19*b ^2-25*b+19)*(1+b)^3/(b-1)^7*n^2+280/9*(1+b)^4/(b-1)^7*n^3 The , 8, -th central moment of an n-evil location for random evil real number\ in base b is 8 7 6 5 4 3 - 2 (43333 b + 675443 b - 73118 b - 1094179 b - 2877422 b - 1094179 b 2 / 8 6 - 73118 b + 675443 b + 43333) / (32805 (b - 1) ) + 2 (1 + b) (352973 b / 5 4 3 2 - 1615146 b - 2336973 b - 2790572 b - 2336973 b - 1615146 b + 352973) / 8 n / (10935 (b - 1) ) / 4 3 2 2 2 4 (77381 b - 36400 b - 79818 b - 36400 b + 77381) (1 + b) n + ---------------------------------------------------------------- 8 1215 (b - 1) 2 3 3 4 4 112 (137 b + 130 b + 137) (1 + b) n 560 (1 + b) n + -------------------------------------- + --------------- 8 8 81 (b - 1) 27 (b - 1) and in Maple notation -2/32805*(43333*b^8+675443*b^7-73118*b^6-1094179*b^5-2877422*b^4-1094179*b^3-\ 73118*b^2+675443*b+43333)/(b-1)^8+2/10935*(1+b)*(352973*b^6-1615146*b^5-2336973 *b^4-2790572*b^3-2336973*b^2-1615146*b+352973)/(b-1)^8*n+4/1215*(77381*b^4-\ 36400*b^3-79818*b^2-36400*b+77381)*(1+b)^2/(b-1)^8*n^2+112/81*(137*b^2+130*b+ 137)*(1+b)^3/(b-1)^8*n^3+560/27*(1+b)^4/(b-1)^8*n^4 The , 9, -th central moment of an n-evil location for random evil real number\ in base b is 8 7 6 5 4 - 2 (1 + b) (8413835 b + 41942263 b + 9727466 b - 42601031 b - 405016138 b 3 2 / 9 - 42601031 b + 9727466 b + 41942263 b + 8413835) / (1082565 (b - 1) ) / 6 5 4 3 2 + 2 (286295 b - 1775274 b - 1973895 b - 3292748 b - 1973895 b 2 / 9 - 1775274 b + 286295) (1 + b) n / (3645 (b - 1) ) / 4 3 2 3 2 8 (16959 b - 22750 b - 10082 b - 22750 b + 16959) (1 + b) n + ---------------------------------------------------------------- 9 135 (b - 1) 2 4 3 5 4 112 (779 b + 10 b + 779) (1 + b) n 2240 (1 + b) n + ------------------------------------- + ---------------- 9 9 81 (b - 1) 9 (b - 1) and in Maple notation -2/1082565*(1+b)*(8413835*b^8+41942263*b^7+9727466*b^6-42601031*b^5-405016138*b ^4-42601031*b^3+9727466*b^2+41942263*b+8413835)/(b-1)^9+2/3645*(286295*b^6-\ 1775274*b^5-1973895*b^4-3292748*b^3-1973895*b^2-1775274*b+286295)*(1+b)^2/(b-1) ^9*n+8/135*(16959*b^4-22750*b^3-10082*b^2-22750*b+16959)*(1+b)^3/(b-1)^9*n^2+ 112/81*(779*b^2+10*b+779)*(1+b)^4/(b-1)^9*n^3+2240/9*(1+b)^5/(b-1)^9*n^4 The , 10, -th central moment of an n-evil location for random evil real numbe\ r in base b is 10 9 8 7 6 - 2 (83460401 b + 327624653 b + 742202421 b + 1049673420 b - 3984525222 b 5 4 3 2 - 8425668786 b - 3984525222 b + 1049673420 b + 742202421 b / 10 8 + 327624653 b + 83460401) / (2525985 (b - 1) ) + 2 (1 + b) (5803417 b / 7 6 5 4 3 - 42893576 b - 197638052 b - 375128248 b - 409761898 b - 375128248 b 2 / 10 - 197638052 b - 42893576 b + 5803417) n / (40095 (b - 1) ) + 4 ( / 6 5 4 3 2 728119 b + 66840 b - 2112591 b - 3454720 b - 2112591 b + 66840 b 2 2 / 10 + 728119) (1 + b) n / (729 (b - 1) ) / 4 3 2 3 3 8 (60611 b + 74060 b + 64482 b + 74060 b + 60611) (1 + b) n + ---------------------------------------------------------------- 10 81 (b - 1) 2 4 4 5 5 1120 (18 b + 25 b + 18) (1 + b) n 1120 (1 + b) n + ------------------------------------ + ---------------- 10 10 9 (b - 1) 9 (b - 1) and in Maple notation -2/2525985*(83460401*b^10+327624653*b^9+742202421*b^8+1049673420*b^7-3984525222 *b^6-8425668786*b^5-3984525222*b^4+1049673420*b^3+742202421*b^2+327624653*b+ 83460401)/(b-1)^10+2/40095*(1+b)*(5803417*b^8-42893576*b^7-197638052*b^6-\ 375128248*b^5-409761898*b^4-375128248*b^3-197638052*b^2-42893576*b+5803417)/(b-\ 1)^10*n+4/729*(728119*b^6+66840*b^5-2112591*b^4-3454720*b^3-2112591*b^2+66840*b +728119)*(1+b)^2/(b-1)^10*n^2+8/81*(60611*b^4+74060*b^3+64482*b^2+74060*b+60611 )*(1+b)^3/(b-1)^10*n^3+1120/9*(18*b^2+25*b+18)*(1+b)^4/(b-1)^10*n^4+1120/9*(1+b )^5/(b-1)^10*n^5 The , 11, -th central moment of an n-evil location for random evil real numbe\ r in base b is 10 9 8 7 - 2 (1 + b) (9038144003 b + 33351940193 b + 75598733943 b + 259566784620 b 6 5 4 3 - 496663553082 b - 1300099340826 b - 496663553082 b + 259566784620 b 2 / 11 + 75598733943 b + 33351940193 b + 9038144003) / (80601885 (b - 1) ) - / 8 7 6 5 22 (11657497 b + 512430548 b + 3758682532 b + 7988754284 b 4 3 2 + 7444470790 b + 7988754284 b + 3758682532 b + 512430548 b + 11657497) 2 / 11 6 5 (1 + b) n / (2066715 (b - 1) ) + 88 (1951793 b - 153825 b / 4 3 2 3 2 - 7348821 b - 12092150 b - 7348821 b - 153825 b + 1951793) (1 + b) n / 11 / (10935 (b - 1) ) / 4 3 2 4 3 616 (39049 b + 25300 b + 25638 b + 25300 b + 39049) (1 + b) n + ------------------------------------------------------------------ 11 729 (b - 1) 2 5 4 6 5 12320 (353 b + 265 b + 353) (1 + b) n 61600 (1 + b) n + ---------------------------------------- + ----------------- 11 11 243 (b - 1) 27 (b - 1) and in Maple notation -2/80601885*(1+b)*(9038144003*b^10+33351940193*b^9+75598733943*b^8+259566784620 *b^7-496663553082*b^6-1300099340826*b^5-496663553082*b^4+259566784620*b^3+ 75598733943*b^2+33351940193*b+9038144003)/(b-1)^11-22/2066715*(11657497*b^8+ 512430548*b^7+3758682532*b^6+7988754284*b^5+7444470790*b^4+7988754284*b^3+ 3758682532*b^2+512430548*b+11657497)*(1+b)^2/(b-1)^11*n+88/10935*(1951793*b^6-\ 153825*b^5-7348821*b^4-12092150*b^3-7348821*b^2-153825*b+1951793)*(1+b)^3/(b-1) ^11*n^2+616/729*(39049*b^4+25300*b^3+25638*b^2+25300*b+39049)*(1+b)^4/(b-1)^11* n^3+12320/243*(353*b^2+265*b+353)*(1+b)^5/(b-1)^11*n^4+61600/27*(1+b)^6/(b-1)^ 11*n^5 The , 12, -th central moment of an n-evil location for random evil real numbe\ r in base b is 12 11 10 9 - 2 (56556847703 b + 666470219685 b + 1573850268780 b + 6571009166849 b 8 7 6 + 5577859180449 b - 28254844511910 b - 56698871835336 b 5 4 3 - 28254844511910 b + 5577859180449 b + 6571009166849 b 2 / + 1573850268780 b + 666470219685 b + 56556847703) / (241805655 / 12 10 9 (b - 1) ) - 2 (1 + b) (76653722963 b + 269640291206 b 8 7 6 + 2688078520503 b + 11040949388040 b + 18881019566070 b 5 4 3 + 20793957435108 b + 18881019566070 b + 11040949388040 b 2 / + 2688078520503 b + 269640291206 b + 76653722963) n / (26867295 / 12 8 7 6 5 (b - 1) ) + 4 (376379081 b + 894435080 b - 1620765508 b - 7685391560 b 4 3 2 - 10906453898 b - 7685391560 b - 1620765508 b + 894435080 b + 376379081 2 2 / 12 6 5 ) (1 + b) n / (25515 (b - 1) ) + 352 (5624623 b + 13176510 b / 4 3 2 3 + 11578929 b + 7556420 b + 11578929 b + 13176510 b + 5624623) (1 + b) 3 / 12 n / (10935 (b - 1) ) / 4 3 2 4 4 176 (186413 b + 410690 b + 488946 b + 410690 b + 186413) (1 + b) n + ----------------------------------------------------------------------- 12 243 (b - 1) 2 5 5 6 6 12320 (21 b + 34 b + 21) (1 + b) n 24640 (1 + b) n + ------------------------------------- + ----------------- 12 12 9 (b - 1) 27 (b - 1) and in Maple notation -2/241805655*(56556847703*b^12+666470219685*b^11+1573850268780*b^10+ 6571009166849*b^9+5577859180449*b^8-28254844511910*b^7-56698871835336*b^6-\ 28254844511910*b^5+5577859180449*b^4+6571009166849*b^3+1573850268780*b^2+ 666470219685*b+56556847703)/(b-1)^12-2/26867295*(1+b)*(76653722963*b^10+ 269640291206*b^9+2688078520503*b^8+11040949388040*b^7+18881019566070*b^6+ 20793957435108*b^5+18881019566070*b^4+11040949388040*b^3+2688078520503*b^2+ 269640291206*b+76653722963)/(b-1)^12*n+4/25515*(376379081*b^8+894435080*b^7-\ 1620765508*b^6-7685391560*b^5-10906453898*b^4-7685391560*b^3-1620765508*b^2+ 894435080*b+376379081)*(1+b)^2/(b-1)^12*n^2+352/10935*(5624623*b^6+13176510*b^5 +11578929*b^4+7556420*b^3+11578929*b^2+13176510*b+5624623)*(1+b)^3/(b-1)^12*n^3 +176/243*(186413*b^4+410690*b^3+488946*b^2+410690*b+186413)*(1+b)^4/(b-1)^12*n^ 4+12320/9*(21*b^2+34*b+21)*(1+b)^5/(b-1)^12*n^5+24640/27*(1+b)^6/(b-1)^12*n^6 The , 13, -th central moment of an n-evil location for random evil real numbe\ r in base b is 12 11 10 2 (1 + b) (12441053539 b - 278187317037 b - 472608293472 b 9 8 7 - 2363662806953 b - 4362206992731 b + 10856293421526 b 6 5 4 + 26308546154160 b + 10856293421526 b - 4362206992731 b 3 2 / - 2363662806953 b - 472608293472 b - 278187317037 b + 12441053539) / ( / 13 10 9 8 18600435 (b - 1) ) - 26 (3472931059 b + 7651860842 b + 61462846391 b 7 6 5 4 + 351749174776 b + 612502305878 b + 632090046268 b + 612502305878 b 3 2 2 + 351749174776 b + 61462846391 b + 7651860842 b + 3472931059) (1 + b) n / 13 8 7 / (2066715 (b - 1) ) + 104 (3897170771 b + 16414473460 b / 6 5 4 3 - 27016865140 b - 145459672820 b - 198481266622 b - 145459672820 b 2 3 2 / - 27016865140 b + 16414473460 b + 3897170771) (1 + b) n / (2066715 / 13 6 5 4 3 (b - 1) ) + 4576 (785239 b + 1870660 b + 1152697 b + 274520 b 2 4 3 / 13 + 1152697 b + 1870660 b + 785239) (1 + b) n / (3645 (b - 1) ) / 4 3 2 5 4 4576 (52447 b + 95235 b + 109084 b + 95235 b + 52447) (1 + b) n + -------------------------------------------------------------------- 13 243 (b - 1) 2 6 5 7 6 32032 (2339 b + 2770 b + 2339) (1 + b) n 640640 (1 + b) n + ------------------------------------------- + ------------------ 13 13 243 (b - 1) 27 (b - 1) and in Maple notation 2/18600435*(1+b)*(12441053539*b^12-278187317037*b^11-472608293472*b^10-\ 2363662806953*b^9-4362206992731*b^8+10856293421526*b^7+26308546154160*b^6+ 10856293421526*b^5-4362206992731*b^4-2363662806953*b^3-472608293472*b^2-\ 278187317037*b+12441053539)/(b-1)^13-26/2066715*(3472931059*b^10+7651860842*b^9 +61462846391*b^8+351749174776*b^7+612502305878*b^6+632090046268*b^5+ 612502305878*b^4+351749174776*b^3+61462846391*b^2+7651860842*b+3472931059)*(1+b )^2/(b-1)^13*n+104/2066715*(3897170771*b^8+16414473460*b^7-27016865140*b^6-\ 145459672820*b^5-198481266622*b^4-145459672820*b^3-27016865140*b^2+16414473460* b+3897170771)*(1+b)^3/(b-1)^13*n^2+4576/3645*(785239*b^6+1870660*b^5+1152697*b^ 4+274520*b^3+1152697*b^2+1870660*b+785239)*(1+b)^4/(b-1)^13*n^3+4576/243*(52447 *b^4+95235*b^3+109084*b^2+95235*b+52447)*(1+b)^5/(b-1)^13*n^4+32032/243*(2339*b ^2+2770*b+2339)*(1+b)^6/(b-1)^13*n^5+640640/27*(1+b)^7/(b-1)^13*n^6 The , 14, -th central moment of an n-evil location for random evil real numbe\ r in base b is 14 13 12 2 (314912475251 b - 1334944913621 b - 7301147470939 b 11 10 9 - 20500522737466 b - 77055501259937 b - 43091012481691 b 8 7 6 + 323520064482441 b + 608711661514068 b + 323520064482441 b 5 4 3 - 43091012481691 b - 77055501259937 b - 20500522737466 b 2 / - 7301147470939 b - 1334944913621 b + 314912475251) / (23914845 / 14 12 11 (b - 1) ) - 2 (1 + b) (973044786367 b + 5082923317908 b 10 9 8 + 17035799898702 b + 125586116752324 b + 407397685823505 b 7 6 5 + 656196940694952 b + 722646802721220 b + 656196940694952 b 4 3 2 + 407397685823505 b + 125586116752324 b + 17035799898702 b / 14 + 5082923317908 b + 973044786367) n / (7971615 (b - 1) ) + 4 ( / 10 9 8 7 96923129785 b + 1382133549472 b + 1043522287029 b - 12619948185312 b 6 5 4 - 36924568025838 b - 49090348464768 b - 36924568025838 b 3 2 - 12619948185312 b + 1043522287029 b + 1382133549472 b + 96923129785) 2 2 / 14 8 7 (1 + b) n / (885735 (b - 1) ) + 104 (4991119891 b + 24470842088 b / 6 5 4 3 + 38395615444 b + 20114296664 b + 2901251506 b + 20114296664 b 2 3 3 / + 38395615444 b + 24470842088 b + 4991119891) (1 + b) n / (98415 / 14 6 5 4 3 (b - 1) ) + 64064 (3631033 b + 13221870 b + 21628599 b + 24102740 b 2 4 4 / 14 + 21628599 b + 13221870 b + 3631033) (1 + b) n / (32805 (b - 1) ) / 4 3 2 5 5 64064 (34555 b + 96964 b + 127410 b + 96964 b + 34555) (1 + b) n + --------------------------------------------------------------------- 14 729 (b - 1) 2 6 6 7 7 448448 (217 b + 380 b + 217) (1 + b) n 640640 (1 + b) n + ----------------------------------------- + ------------------ 14 14 243 (b - 1) 81 (b - 1) and in Maple notation 2/23914845*(314912475251*b^14-1334944913621*b^13-7301147470939*b^12-\ 20500522737466*b^11-77055501259937*b^10-43091012481691*b^9+323520064482441*b^8+ 608711661514068*b^7+323520064482441*b^6-43091012481691*b^5-77055501259937*b^4-\ 20500522737466*b^3-7301147470939*b^2-1334944913621*b+314912475251)/(b-1)^14-2/ 7971615*(1+b)*(973044786367*b^12+5082923317908*b^11+17035799898702*b^10+ 125586116752324*b^9+407397685823505*b^8+656196940694952*b^7+722646802721220*b^6 +656196940694952*b^5+407397685823505*b^4+125586116752324*b^3+17035799898702*b^2 +5082923317908*b+973044786367)/(b-1)^14*n+4/885735*(96923129785*b^10+ 1382133549472*b^9+1043522287029*b^8-12619948185312*b^7-36924568025838*b^6-\ 49090348464768*b^5-36924568025838*b^4-12619948185312*b^3+1043522287029*b^2+ 1382133549472*b+96923129785)*(1+b)^2/(b-1)^14*n^2+104/98415*(4991119891*b^8+ 24470842088*b^7+38395615444*b^6+20114296664*b^5+2901251506*b^4+20114296664*b^3+ 38395615444*b^2+24470842088*b+4991119891)*(1+b)^3/(b-1)^14*n^3+64064/32805*( 3631033*b^6+13221870*b^5+21628599*b^4+24102740*b^3+21628599*b^2+13221870*b+ 3631033)*(1+b)^4/(b-1)^14*n^4+64064/729*(34555*b^4+96964*b^3+127410*b^2+96964*b +34555)*(1+b)^5/(b-1)^14*n^5+448448/243*(217*b^2+380*b+217)*(1+b)^6/(b-1)^14*n^ 6+640640/81*(1+b)^7/(b-1)^14*n^7 The , 15, -th central moment of an n-evil location for random evil real numbe\ r in base b is 14 13 12 2 (1 + b) (4862702300619701 b - 5080572043802621 b - 92425168211792749 b 11 10 9 - 155700548061367546 b - 852227219512047479 b - 1069602147282555955 b 8 7 6 + 3566619005983867695 b + 7717811781203360340 b + 3566619005983867695 b 5 4 3 - 1069602147282555955 b - 852227219512047479 b - 155700548061367546 b 2 / - 92425168211792749 b - 5080572043802621 b + 4862702300619701) / ( / 15 12 11 42687998325 (b - 1) ) - 2 (4132897047839 b + 39492064830840 b 10 9 8 + 77522711775126 b + 665321944020056 b + 2636078524261137 b 7 6 5 + 4270861143605808 b + 4498557028132980 b + 4270861143605808 b 4 3 2 + 2636078524261137 b + 665321944020056 b + 77522711775126 b 2 / 15 + 39492064830840 b + 4132897047839) (1 + b) n / (7971615 (b - 1) ) - 8 / 10 9 8 7 (33607281571 b - 1175738402605 b - 1902081492097 b + 13703872585780 b 6 5 4 + 42716204657054 b + 56149129286770 b + 42716204657054 b 3 2 + 13703872585780 b - 1902081492097 b - 1175738402605 b + 33607281571) 3 2 / 15 8 (1 + b) n / (295245 (b - 1) ) + 104 (231774050381 b / 7 6 5 4 + 1433983575640 b + 2217546807404 b + 509244578920 b - 807976013554 b 3 2 + 509244578920 b + 2217546807404 b + 1433983575640 b + 231774050381) 4 3 / 15 6 5 (1 + b) n / (885735 (b - 1) ) + 64064 (5140207 b + 19216095 b / 4 3 2 5 + 29007681 b + 30769490 b + 29007681 b + 19216095 b + 5140207) (1 + b) 4 / 15 n / (6561 (b - 1) ) / 4 3 2 6 5 320320 (21577 b + 54978 b + 69394 b + 54978 b + 21577) (1 + b) n + --------------------------------------------------------------------- 15 243 (b - 1) 2 7 6 8 7 22422400 (181 b + 263 b + 181) (1 + b) n 22422400 (1 + b) n + ------------------------------------------- + -------------------- 15 15 729 (b - 1) 81 (b - 1) and in Maple notation 2/42687998325*(1+b)*(4862702300619701*b^14-5080572043802621*b^13-\ 92425168211792749*b^12-155700548061367546*b^11-852227219512047479*b^10-\ 1069602147282555955*b^9+3566619005983867695*b^8+7717811781203360340*b^7+ 3566619005983867695*b^6-1069602147282555955*b^5-852227219512047479*b^4-\ 155700548061367546*b^3-92425168211792749*b^2-5080572043802621*b+ 4862702300619701)/(b-1)^15-2/7971615*(4132897047839*b^12+39492064830840*b^11+ 77522711775126*b^10+665321944020056*b^9+2636078524261137*b^8+4270861143605808*b ^7+4498557028132980*b^6+4270861143605808*b^5+2636078524261137*b^4+ 665321944020056*b^3+77522711775126*b^2+39492064830840*b+4132897047839)*(1+b)^2/ (b-1)^15*n-8/295245*(33607281571*b^10-1175738402605*b^9-1902081492097*b^8+ 13703872585780*b^7+42716204657054*b^6+56149129286770*b^5+42716204657054*b^4+ 13703872585780*b^3-1902081492097*b^2-1175738402605*b+33607281571)*(1+b)^3/(b-1) ^15*n^2+104/885735*(231774050381*b^8+1433983575640*b^7+2217546807404*b^6+ 509244578920*b^5-807976013554*b^4+509244578920*b^3+2217546807404*b^2+ 1433983575640*b+231774050381)*(1+b)^4/(b-1)^15*n^3+64064/6561*(5140207*b^6+ 19216095*b^5+29007681*b^4+30769490*b^3+29007681*b^2+19216095*b+5140207)*(1+b)^5 /(b-1)^15*n^4+320320/243*(21577*b^4+54978*b^3+69394*b^2+54978*b+21577)*(1+b)^6/ (b-1)^15*n^5+22422400/729*(181*b^2+263*b+181)*(1+b)^7/(b-1)^15*n^6+22422400/81* (1+b)^8/(b-1)^15*n^7 The , 16, -th central moment of an n-evil location for random evil real numbe\ r in base b is 16 15 14 2 (6449832315034223 b + 25488004094654753 b - 118764817851305310 b 13 12 - 429906530029624421 b - 1389776469778337144 b 11 10 - 4507515527051552919 b - 1294539391368570562 b 9 8 + 18919668382786452443 b + 33608870232999128658 b 7 6 + 18919668382786452443 b - 1294539391368570562 b 5 4 3 - 4507515527051552919 b - 1389776469778337144 b - 429906530029624421 b 2 / - 118764817851305310 b + 25488004094654753 b + 6449832315034223) / ( / 16 14 8537599665 (b - 1) ) - 2 (1 + b) (987632706646077 b 13 12 11 + 38862356163131582 b + 130388386703198079 b + 545177480837683692 b 10 9 + 3052643241610719509 b + 8443564213287135138 b 8 7 + 12987244799027812143 b + 14287208359294143336 b 6 5 + 12987244799027812143 b + 8443564213287135138 b 4 3 2 + 3052643241610719509 b + 545177480837683692 b + 130388386703198079 b / 16 + 38862356163131582 b + 987632706646077) n / (948622185 (b - 1) ) - 4 ( / 12 11 10 43810661897147 b - 185355491962080 b - 1610325377171346 b 9 8 7 + 2477696469665120 b + 26251599534660405 b + 61835010077554560 b 6 5 4 + 78875189721956676 b + 61835010077554560 b + 26251599534660405 b 3 2 + 2477696469665120 b - 1610325377171346 b - 185355491962080 b 2 2 / 16 + 43810661897147) (1 + b) n / (7971615 (b - 1) ) + 16 ( / 10 9 8 7221322318051 b + 79043527232254 b + 242475276123207 b 7 6 5 + 267017058722856 b + 8880691292886 b - 174474639633996 b 4 3 2 + 8880691292886 b + 267017058722856 b + 242475276123207 b 3 3 / 16 + 79043527232254 b + 7221322318051) (1 + b) n / (885735 (b - 1) ) + / 8 7 6 5 208 (55199003297 b + 341713351980 b + 833784795908 b + 1177670803540 b 4 3 2 + 1262901725622 b + 1177670803540 b + 833784795908 b + 341713351980 b 4 4 / 16 6 + 55199003297) (1 + b) n / (32805 (b - 1) ) + 64064 (26419163 b / 5 4 3 2 + 117761898 b + 232185189 b + 282058156 b + 232185189 b + 117761898 b 5 5 / 16 + 26419163) (1 + b) n / (6561 (b - 1) ) / 4 3 2 6 6 640640 (26351 b + 84168 b + 116210 b + 84168 b + 26351) (1 + b) n + ---------------------------------------------------------------------- 16 243 (b - 1) 2 7 7 8 8 2562560 (191 b + 350 b + 191) (1 + b) n 6406400 (1 + b) n + ------------------------------------------ + ------------------- 16 16 81 (b - 1) 81 (b - 1) and in Maple notation 2/8537599665*(6449832315034223*b^16+25488004094654753*b^15-118764817851305310*b ^14-429906530029624421*b^13-1389776469778337144*b^12-4507515527051552919*b^11-\ 1294539391368570562*b^10+18919668382786452443*b^9+33608870232999128658*b^8+ 18919668382786452443*b^7-1294539391368570562*b^6-4507515527051552919*b^5-\ 1389776469778337144*b^4-429906530029624421*b^3-118764817851305310*b^2+ 25488004094654753*b+6449832315034223)/(b-1)^16-2/948622185*(1+b)*( 987632706646077*b^14+38862356163131582*b^13+130388386703198079*b^12+ 545177480837683692*b^11+3052643241610719509*b^10+8443564213287135138*b^9+ 12987244799027812143*b^8+14287208359294143336*b^7+12987244799027812143*b^6+ 8443564213287135138*b^5+3052643241610719509*b^4+545177480837683692*b^3+ 130388386703198079*b^2+38862356163131582*b+987632706646077)/(b-1)^16*n-4/ 7971615*(43810661897147*b^12-185355491962080*b^11-1610325377171346*b^10+ 2477696469665120*b^9+26251599534660405*b^8+61835010077554560*b^7+ 78875189721956676*b^6+61835010077554560*b^5+26251599534660405*b^4+ 2477696469665120*b^3-1610325377171346*b^2-185355491962080*b+43810661897147)*(1+ b)^2/(b-1)^16*n^2+16/885735*(7221322318051*b^10+79043527232254*b^9+ 242475276123207*b^8+267017058722856*b^7+8880691292886*b^6-174474639633996*b^5+ 8880691292886*b^4+267017058722856*b^3+242475276123207*b^2+79043527232254*b+ 7221322318051)*(1+b)^3/(b-1)^16*n^3+208/32805*(55199003297*b^8+341713351980*b^7 +833784795908*b^6+1177670803540*b^5+1262901725622*b^4+1177670803540*b^3+ 833784795908*b^2+341713351980*b+55199003297)*(1+b)^4/(b-1)^16*n^4+64064/6561*( 26419163*b^6+117761898*b^5+232185189*b^4+282058156*b^3+232185189*b^2+117761898* b+26419163)*(1+b)^5/(b-1)^16*n^5+640640/243*(26351*b^4+84168*b^3+116210*b^2+ 84168*b+26351)*(1+b)^6/(b-1)^16*n^6+2562560/81*(191*b^2+350*b+191)*(1+b)^7/(b-1 )^16*n^7+6406400/81*(1+b)^8/(b-1)^16*n^8 To sum up the asymptotics, up to exponentially decaying terms for the first , 16, central moments are 2 -5 + b 2 n 2 (b - 7 b + 1) 2 (1 + b) n [--------- + -----, ---------------- + -----------, 3 (b - 1) b - 1 2 2 9 (b - 1) 3 (b - 1) 2 2 2 (1 + b) (11 b - 95 b + 11) 2 (1 + b) n ----------------------------- + ------------, 3 3 135 (b - 1) 3 (b - 1) 2 2 2 2 (17 b - 11 b + 17) (b - 10 b + 1) 2 (1 + b) (13 b - 30 b + 13) n ------------------------------------- + ------------------------------- 4 4 135 (b - 1) 15 (b - 1) 2 2 4 3 2 4 (1 + b) n 2 (1 + b) (293 b - 4291 b + 7356 b - 4291 b + 293) + -------------, ----------------------------------------------------- 4 5 3 (b - 1) 1701 (b - 1) 2 2 3 2 10 (31 b - 118 b + 31) (1 + b) n 40 (1 + b) n + ---------------------------------- + --------------, 2 5 5 81 (b - 1) 9 (b - 1) 6 5 4 3 2 (5413 b - 120459 b + 116799 b + 77102 b + 116799 b - 120459 b + 5413) / 6 / (25515 (b - 1) ) / 4 3 2 2 (1 + b) (2713 b - 9884 b - 6618 b - 9884 b + 2713) n + --------------------------------------------------------- 6 567 (b - 1) 2 2 2 3 3 52 (b + 1) (1 + b) n 40 (1 + b) n + ----------------------- + --------------, 2 (1 + b) 6 6 3 (b - 1) 9 (b - 1) 6 5 4 3 2 / (125 b - 35517 b + 61143 b - 12494 b + 61143 b - 35517 b + 125) / ( / 4 3 2 2 7 14 (717 b - 3680 b + 854 b - 3680 b + 717) (1 + b) n 3645 (b - 1) ) + -------------------------------------------------------- 7 405 (b - 1) 2 3 2 4 3 280 (19 b - 25 b + 19) (1 + b) n 280 (1 + b) n 8 + ----------------------------------- + ---------------, - 2 (43333 b 7 7 81 (b - 1) 9 (b - 1) 7 6 5 4 3 2 + 675443 b - 73118 b - 1094179 b - 2877422 b - 1094179 b - 73118 b / 8 6 + 675443 b + 43333) / (32805 (b - 1) ) + 2 (1 + b) (352973 b / 5 4 3 2 - 1615146 b - 2336973 b - 2790572 b - 2336973 b - 1615146 b + 352973) / 8 n / (10935 (b - 1) ) / 4 3 2 2 2 4 (77381 b - 36400 b - 79818 b - 36400 b + 77381) (1 + b) n + ---------------------------------------------------------------- 8 1215 (b - 1) 2 3 3 4 4 112 (137 b + 130 b + 137) (1 + b) n 560 (1 + b) n + -------------------------------------- + ---------------, - 2 (1 + b) ( 8 8 81 (b - 1) 27 (b - 1) 8 7 6 5 4 8413835 b + 41942263 b + 9727466 b - 42601031 b - 405016138 b 3 2 / 9 - 42601031 b + 9727466 b + 41942263 b + 8413835) / (1082565 (b - 1) ) / 6 5 4 3 2 + 2 (286295 b - 1775274 b - 1973895 b - 3292748 b - 1973895 b 2 / 9 - 1775274 b + 286295) (1 + b) n / (3645 (b - 1) ) / 4 3 2 3 2 8 (16959 b - 22750 b - 10082 b - 22750 b + 16959) (1 + b) n + ---------------------------------------------------------------- 9 135 (b - 1) 2 4 3 5 4 112 (779 b + 10 b + 779) (1 + b) n 2240 (1 + b) n + ------------------------------------- + ----------------, - 2 ( 9 9 81 (b - 1) 9 (b - 1) 10 9 8 7 6 83460401 b + 327624653 b + 742202421 b + 1049673420 b - 3984525222 b 5 4 3 2 - 8425668786 b - 3984525222 b + 1049673420 b + 742202421 b / 10 8 + 327624653 b + 83460401) / (2525985 (b - 1) ) + 2 (1 + b) (5803417 b / 7 6 5 4 3 - 42893576 b - 197638052 b - 375128248 b - 409761898 b - 375128248 b 2 / 10 - 197638052 b - 42893576 b + 5803417) n / (40095 (b - 1) ) + 4 ( / 6 5 4 3 2 728119 b + 66840 b - 2112591 b - 3454720 b - 2112591 b + 66840 b 2 2 / 10 + 728119) (1 + b) n / (729 (b - 1) ) / 4 3 2 3 3 8 (60611 b + 74060 b + 64482 b + 74060 b + 60611) (1 + b) n + ---------------------------------------------------------------- 10 81 (b - 1) 2 4 4 5 5 1120 (18 b + 25 b + 18) (1 + b) n 1120 (1 + b) n + ------------------------------------ + ----------------, - 2 (1 + b) ( 10 10 9 (b - 1) 9 (b - 1) 10 9 8 7 9038144003 b + 33351940193 b + 75598733943 b + 259566784620 b 6 5 4 3 - 496663553082 b - 1300099340826 b - 496663553082 b + 259566784620 b 2 / 11 + 75598733943 b + 33351940193 b + 9038144003) / (80601885 (b - 1) ) - / 8 7 6 5 22 (11657497 b + 512430548 b + 3758682532 b + 7988754284 b 4 3 2 + 7444470790 b + 7988754284 b + 3758682532 b + 512430548 b + 11657497) 2 / 11 6 5 (1 + b) n / (2066715 (b - 1) ) + 88 (1951793 b - 153825 b / 4 3 2 3 2 - 7348821 b - 12092150 b - 7348821 b - 153825 b + 1951793) (1 + b) n / 11 / (10935 (b - 1) ) / 4 3 2 4 3 616 (39049 b + 25300 b + 25638 b + 25300 b + 39049) (1 + b) n + ------------------------------------------------------------------ 11 729 (b - 1) 2 5 4 6 5 12320 (353 b + 265 b + 353) (1 + b) n 61600 (1 + b) n + ---------------------------------------- + -----------------, - 2 ( 11 11 243 (b - 1) 27 (b - 1) 12 11 10 9 56556847703 b + 666470219685 b + 1573850268780 b + 6571009166849 b 8 7 6 + 5577859180449 b - 28254844511910 b - 56698871835336 b 5 4 3 - 28254844511910 b + 5577859180449 b + 6571009166849 b 2 / + 1573850268780 b + 666470219685 b + 56556847703) / (241805655 / 12 10 9 (b - 1) ) - 2 (1 + b) (76653722963 b + 269640291206 b 8 7 6 + 2688078520503 b + 11040949388040 b + 18881019566070 b 5 4 3 + 20793957435108 b + 18881019566070 b + 11040949388040 b 2 / + 2688078520503 b + 269640291206 b + 76653722963) n / (26867295 / 12 8 7 6 5 (b - 1) ) + 4 (376379081 b + 894435080 b - 1620765508 b - 7685391560 b 4 3 2 - 10906453898 b - 7685391560 b - 1620765508 b + 894435080 b + 376379081 2 2 / 12 6 5 ) (1 + b) n / (25515 (b - 1) ) + 352 (5624623 b + 13176510 b / 4 3 2 3 + 11578929 b + 7556420 b + 11578929 b + 13176510 b + 5624623) (1 + b) 3 / 12 n / (10935 (b - 1) ) / 4 3 2 4 4 176 (186413 b + 410690 b + 488946 b + 410690 b + 186413) (1 + b) n + ----------------------------------------------------------------------- 12 243 (b - 1) 2 5 5 6 6 12320 (21 b + 34 b + 21) (1 + b) n 24640 (1 + b) n + ------------------------------------- + -----------------, 2 (1 + b) ( 12 12 9 (b - 1) 27 (b - 1) 12 11 10 9 12441053539 b - 278187317037 b - 472608293472 b - 2363662806953 b 8 7 6 - 4362206992731 b + 10856293421526 b + 26308546154160 b 5 4 3 + 10856293421526 b - 4362206992731 b - 2363662806953 b 2 / 13 - 472608293472 b - 278187317037 b + 12441053539) / (18600435 (b - 1) ) / 10 9 8 7 - 26 (3472931059 b + 7651860842 b + 61462846391 b + 351749174776 b 6 5 4 3 + 612502305878 b + 632090046268 b + 612502305878 b + 351749174776 b 2 2 / + 61462846391 b + 7651860842 b + 3472931059) (1 + b) n / (2066715 / 13 8 7 6 (b - 1) ) + 104 (3897170771 b + 16414473460 b - 27016865140 b 5 4 3 2 - 145459672820 b - 198481266622 b - 145459672820 b - 27016865140 b 3 2 / 13 + 16414473460 b + 3897170771) (1 + b) n / (2066715 (b - 1) ) + 4576 ( / 6 5 4 3 2 785239 b + 1870660 b + 1152697 b + 274520 b + 1152697 b + 1870660 b 4 3 / 13 + 785239) (1 + b) n / (3645 (b - 1) ) / 4 3 2 5 4 4576 (52447 b + 95235 b + 109084 b + 95235 b + 52447) (1 + b) n + -------------------------------------------------------------------- 13 243 (b - 1) 2 6 5 7 6 32032 (2339 b + 2770 b + 2339) (1 + b) n 640640 (1 + b) n + ------------------------------------------- + ------------------, 2 ( 13 13 243 (b - 1) 27 (b - 1) 14 13 12 314912475251 b - 1334944913621 b - 7301147470939 b 11 10 9 - 20500522737466 b - 77055501259937 b - 43091012481691 b 8 7 6 + 323520064482441 b + 608711661514068 b + 323520064482441 b 5 4 3 - 43091012481691 b - 77055501259937 b - 20500522737466 b 2 / - 7301147470939 b - 1334944913621 b + 314912475251) / (23914845 / 14 12 11 (b - 1) ) - 2 (1 + b) (973044786367 b + 5082923317908 b 10 9 8 + 17035799898702 b + 125586116752324 b + 407397685823505 b 7 6 5 + 656196940694952 b + 722646802721220 b + 656196940694952 b 4 3 2 + 407397685823505 b + 125586116752324 b + 17035799898702 b / 14 + 5082923317908 b + 973044786367) n / (7971615 (b - 1) ) + 4 ( / 10 9 8 7 96923129785 b + 1382133549472 b + 1043522287029 b - 12619948185312 b 6 5 4 - 36924568025838 b - 49090348464768 b - 36924568025838 b 3 2 - 12619948185312 b + 1043522287029 b + 1382133549472 b + 96923129785) 2 2 / 14 8 7 (1 + b) n / (885735 (b - 1) ) + 104 (4991119891 b + 24470842088 b / 6 5 4 3 + 38395615444 b + 20114296664 b + 2901251506 b + 20114296664 b 2 3 3 / + 38395615444 b + 24470842088 b + 4991119891) (1 + b) n / (98415 / 14 6 5 4 3 (b - 1) ) + 64064 (3631033 b + 13221870 b + 21628599 b + 24102740 b 2 4 4 / 14 + 21628599 b + 13221870 b + 3631033) (1 + b) n / (32805 (b - 1) ) / 4 3 2 5 5 64064 (34555 b + 96964 b + 127410 b + 96964 b + 34555) (1 + b) n + --------------------------------------------------------------------- 14 729 (b - 1) 2 6 6 7 7 448448 (217 b + 380 b + 217) (1 + b) n 640640 (1 + b) n + ----------------------------------------- + ------------------, 2 14 14 243 (b - 1) 81 (b - 1) 14 13 (1 + b) (4862702300619701 b - 5080572043802621 b 12 11 10 - 92425168211792749 b - 155700548061367546 b - 852227219512047479 b 9 8 7 - 1069602147282555955 b + 3566619005983867695 b + 7717811781203360340 b 6 5 4 + 3566619005983867695 b - 1069602147282555955 b - 852227219512047479 b 3 2 - 155700548061367546 b - 92425168211792749 b - 5080572043802621 b / 15 12 + 4862702300619701) / (42687998325 (b - 1) ) - 2 (4132897047839 b / 11 10 9 + 39492064830840 b + 77522711775126 b + 665321944020056 b 8 7 6 + 2636078524261137 b + 4270861143605808 b + 4498557028132980 b 5 4 3 + 4270861143605808 b + 2636078524261137 b + 665321944020056 b 2 2 / + 77522711775126 b + 39492064830840 b + 4132897047839) (1 + b) n / ( / 15 10 9 7971615 (b - 1) ) - 8 (33607281571 b - 1175738402605 b 8 7 6 - 1902081492097 b + 13703872585780 b + 42716204657054 b 5 4 3 + 56149129286770 b + 42716204657054 b + 13703872585780 b 2 3 2 / - 1902081492097 b - 1175738402605 b + 33607281571) (1 + b) n / ( / 15 8 7 295245 (b - 1) ) + 104 (231774050381 b + 1433983575640 b 6 5 4 3 + 2217546807404 b + 509244578920 b - 807976013554 b + 509244578920 b 2 4 3 / + 2217546807404 b + 1433983575640 b + 231774050381) (1 + b) n / ( / 15 6 5 4 885735 (b - 1) ) + 64064 (5140207 b + 19216095 b + 29007681 b 3 2 5 4 / + 30769490 b + 29007681 b + 19216095 b + 5140207) (1 + b) n / (6561 / 15 (b - 1) ) 4 3 2 6 5 320320 (21577 b + 54978 b + 69394 b + 54978 b + 21577) (1 + b) n + --------------------------------------------------------------------- 15 243 (b - 1) 2 7 6 8 7 22422400 (181 b + 263 b + 181) (1 + b) n 22422400 (1 + b) n + ------------------------------------------- + --------------------, 2 ( 15 15 729 (b - 1) 81 (b - 1) 16 15 14 6449832315034223 b + 25488004094654753 b - 118764817851305310 b 13 12 - 429906530029624421 b - 1389776469778337144 b 11 10 - 4507515527051552919 b - 1294539391368570562 b 9 8 + 18919668382786452443 b + 33608870232999128658 b 7 6 + 18919668382786452443 b - 1294539391368570562 b 5 4 3 - 4507515527051552919 b - 1389776469778337144 b - 429906530029624421 b 2 / - 118764817851305310 b + 25488004094654753 b + 6449832315034223) / ( / 16 14 8537599665 (b - 1) ) - 2 (1 + b) (987632706646077 b 13 12 11 + 38862356163131582 b + 130388386703198079 b + 545177480837683692 b 10 9 + 3052643241610719509 b + 8443564213287135138 b 8 7 + 12987244799027812143 b + 14287208359294143336 b 6 5 + 12987244799027812143 b + 8443564213287135138 b 4 3 2 + 3052643241610719509 b + 545177480837683692 b + 130388386703198079 b / 16 + 38862356163131582 b + 987632706646077) n / (948622185 (b - 1) ) - 4 ( / 12 11 10 43810661897147 b - 185355491962080 b - 1610325377171346 b 9 8 7 + 2477696469665120 b + 26251599534660405 b + 61835010077554560 b 6 5 4 + 78875189721956676 b + 61835010077554560 b + 26251599534660405 b 3 2 + 2477696469665120 b - 1610325377171346 b - 185355491962080 b 2 2 / 16 + 43810661897147) (1 + b) n / (7971615 (b - 1) ) + 16 ( / 10 9 8 7221322318051 b + 79043527232254 b + 242475276123207 b 7 6 5 + 267017058722856 b + 8880691292886 b - 174474639633996 b 4 3 2 + 8880691292886 b + 267017058722856 b + 242475276123207 b 3 3 / 16 + 79043527232254 b + 7221322318051) (1 + b) n / (885735 (b - 1) ) + / 8 7 6 5 208 (55199003297 b + 341713351980 b + 833784795908 b + 1177670803540 b 4 3 2 + 1262901725622 b + 1177670803540 b + 833784795908 b + 341713351980 b 4 4 / 16 6 + 55199003297) (1 + b) n / (32805 (b - 1) ) + 64064 (26419163 b / 5 4 3 2 + 117761898 b + 232185189 b + 282058156 b + 232185189 b + 117761898 b 5 5 / 16 + 26419163) (1 + b) n / (6561 (b - 1) ) / 4 3 2 6 6 640640 (26351 b + 84168 b + 116210 b + 84168 b + 26351) (1 + b) n + ---------------------------------------------------------------------- 16 243 (b - 1) 2 7 7 8 8 2562560 (191 b + 350 b + 191) (1 + b) n 6406400 (1 + b) n + ------------------------------------------ + -------------------] 16 16 81 (b - 1) 81 (b - 1) and in Maple notation [1/3*(-5+b)/(b-1)+2/(b-1)*n, 2/9*(b^2-7*b+1)/(b-1)^2+2/3*(1+b)/(b-1)^2*n, 2/135 *(1+b)*(11*b^2-95*b+11)/(b-1)^3+2/3*(1+b)^2/(b-1)^3*n, 2/135*(17*b^2-11*b+17)*( b^2-10*b+1)/(b-1)^4+2/15*(1+b)*(13*b^2-30*b+13)/(b-1)^4*n+4/3*(1+b)^2/(b-1)^4*n ^2, 2/1701*(1+b)*(293*b^4-4291*b^3+7356*b^2-4291*b+293)/(b-1)^5+10/81*(31*b^2-\ 118*b+31)*(1+b)^2/(b-1)^5*n+40/9*(1+b)^3/(b-1)^5*n^2, 2/25515*(5413*b^6-120459* b^5+116799*b^4+77102*b^3+116799*b^2-120459*b+5413)/(b-1)^6+2/567*(1+b)*(2713*b^ 4-9884*b^3-6618*b^2-9884*b+2713)/(b-1)^6*n+52/3*(b^2+1)*(1+b)^2/(b-1)^6*n^2+40/ 9*(1+b)^3/(b-1)^6*n^3, 2/3645*(1+b)*(125*b^6-35517*b^5+61143*b^4-12494*b^3+ 61143*b^2-35517*b+125)/(b-1)^7+14/405*(717*b^4-3680*b^3+854*b^2-3680*b+717)*(1+ b)^2/(b-1)^7*n+280/81*(19*b^2-25*b+19)*(1+b)^3/(b-1)^7*n^2+280/9*(1+b)^4/(b-1)^ 7*n^3, -2/32805*(43333*b^8+675443*b^7-73118*b^6-1094179*b^5-2877422*b^4-1094179 *b^3-73118*b^2+675443*b+43333)/(b-1)^8+2/10935*(1+b)*(352973*b^6-1615146*b^5-\ 2336973*b^4-2790572*b^3-2336973*b^2-1615146*b+352973)/(b-1)^8*n+4/1215*(77381*b ^4-36400*b^3-79818*b^2-36400*b+77381)*(1+b)^2/(b-1)^8*n^2+112/81*(137*b^2+130*b +137)*(1+b)^3/(b-1)^8*n^3+560/27*(1+b)^4/(b-1)^8*n^4, -2/1082565*(1+b)*(8413835 *b^8+41942263*b^7+9727466*b^6-42601031*b^5-405016138*b^4-42601031*b^3+9727466*b ^2+41942263*b+8413835)/(b-1)^9+2/3645*(286295*b^6-1775274*b^5-1973895*b^4-\ 3292748*b^3-1973895*b^2-1775274*b+286295)*(1+b)^2/(b-1)^9*n+8/135*(16959*b^4-\ 22750*b^3-10082*b^2-22750*b+16959)*(1+b)^3/(b-1)^9*n^2+112/81*(779*b^2+10*b+779 )*(1+b)^4/(b-1)^9*n^3+2240/9*(1+b)^5/(b-1)^9*n^4, -2/2525985*(83460401*b^10+ 327624653*b^9+742202421*b^8+1049673420*b^7-3984525222*b^6-8425668786*b^5-\ 3984525222*b^4+1049673420*b^3+742202421*b^2+327624653*b+83460401)/(b-1)^10+2/ 40095*(1+b)*(5803417*b^8-42893576*b^7-197638052*b^6-375128248*b^5-409761898*b^4 -375128248*b^3-197638052*b^2-42893576*b+5803417)/(b-1)^10*n+4/729*(728119*b^6+ 66840*b^5-2112591*b^4-3454720*b^3-2112591*b^2+66840*b+728119)*(1+b)^2/(b-1)^10* n^2+8/81*(60611*b^4+74060*b^3+64482*b^2+74060*b+60611)*(1+b)^3/(b-1)^10*n^3+ 1120/9*(18*b^2+25*b+18)*(1+b)^4/(b-1)^10*n^4+1120/9*(1+b)^5/(b-1)^10*n^5, -2/ 80601885*(1+b)*(9038144003*b^10+33351940193*b^9+75598733943*b^8+259566784620*b^ 7-496663553082*b^6-1300099340826*b^5-496663553082*b^4+259566784620*b^3+ 75598733943*b^2+33351940193*b+9038144003)/(b-1)^11-22/2066715*(11657497*b^8+ 512430548*b^7+3758682532*b^6+7988754284*b^5+7444470790*b^4+7988754284*b^3+ 3758682532*b^2+512430548*b+11657497)*(1+b)^2/(b-1)^11*n+88/10935*(1951793*b^6-\ 153825*b^5-7348821*b^4-12092150*b^3-7348821*b^2-153825*b+1951793)*(1+b)^3/(b-1) ^11*n^2+616/729*(39049*b^4+25300*b^3+25638*b^2+25300*b+39049)*(1+b)^4/(b-1)^11* n^3+12320/243*(353*b^2+265*b+353)*(1+b)^5/(b-1)^11*n^4+61600/27*(1+b)^6/(b-1)^ 11*n^5, -2/241805655*(56556847703*b^12+666470219685*b^11+1573850268780*b^10+ 6571009166849*b^9+5577859180449*b^8-28254844511910*b^7-56698871835336*b^6-\ 28254844511910*b^5+5577859180449*b^4+6571009166849*b^3+1573850268780*b^2+ 666470219685*b+56556847703)/(b-1)^12-2/26867295*(1+b)*(76653722963*b^10+ 269640291206*b^9+2688078520503*b^8+11040949388040*b^7+18881019566070*b^6+ 20793957435108*b^5+18881019566070*b^4+11040949388040*b^3+2688078520503*b^2+ 269640291206*b+76653722963)/(b-1)^12*n+4/25515*(376379081*b^8+894435080*b^7-\ 1620765508*b^6-7685391560*b^5-10906453898*b^4-7685391560*b^3-1620765508*b^2+ 894435080*b+376379081)*(1+b)^2/(b-1)^12*n^2+352/10935*(5624623*b^6+13176510*b^5 +11578929*b^4+7556420*b^3+11578929*b^2+13176510*b+5624623)*(1+b)^3/(b-1)^12*n^3 +176/243*(186413*b^4+410690*b^3+488946*b^2+410690*b+186413)*(1+b)^4/(b-1)^12*n^ 4+12320/9*(21*b^2+34*b+21)*(1+b)^5/(b-1)^12*n^5+24640/27*(1+b)^6/(b-1)^12*n^6, 2/18600435*(1+b)*(12441053539*b^12-278187317037*b^11-472608293472*b^10-\ 2363662806953*b^9-4362206992731*b^8+10856293421526*b^7+26308546154160*b^6+ 10856293421526*b^5-4362206992731*b^4-2363662806953*b^3-472608293472*b^2-\ 278187317037*b+12441053539)/(b-1)^13-26/2066715*(3472931059*b^10+7651860842*b^9 +61462846391*b^8+351749174776*b^7+612502305878*b^6+632090046268*b^5+ 612502305878*b^4+351749174776*b^3+61462846391*b^2+7651860842*b+3472931059)*(1+b )^2/(b-1)^13*n+104/2066715*(3897170771*b^8+16414473460*b^7-27016865140*b^6-\ 145459672820*b^5-198481266622*b^4-145459672820*b^3-27016865140*b^2+16414473460* b+3897170771)*(1+b)^3/(b-1)^13*n^2+4576/3645*(785239*b^6+1870660*b^5+1152697*b^ 4+274520*b^3+1152697*b^2+1870660*b+785239)*(1+b)^4/(b-1)^13*n^3+4576/243*(52447 *b^4+95235*b^3+109084*b^2+95235*b+52447)*(1+b)^5/(b-1)^13*n^4+32032/243*(2339*b ^2+2770*b+2339)*(1+b)^6/(b-1)^13*n^5+640640/27*(1+b)^7/(b-1)^13*n^6, 2/23914845 *(314912475251*b^14-1334944913621*b^13-7301147470939*b^12-20500522737466*b^11-\ 77055501259937*b^10-43091012481691*b^9+323520064482441*b^8+608711661514068*b^7+ 323520064482441*b^6-43091012481691*b^5-77055501259937*b^4-20500522737466*b^3-\ 7301147470939*b^2-1334944913621*b+314912475251)/(b-1)^14-2/7971615*(1+b)*( 973044786367*b^12+5082923317908*b^11+17035799898702*b^10+125586116752324*b^9+ 407397685823505*b^8+656196940694952*b^7+722646802721220*b^6+656196940694952*b^5 +407397685823505*b^4+125586116752324*b^3+17035799898702*b^2+5082923317908*b+ 973044786367)/(b-1)^14*n+4/885735*(96923129785*b^10+1382133549472*b^9+ 1043522287029*b^8-12619948185312*b^7-36924568025838*b^6-49090348464768*b^5-\ 36924568025838*b^4-12619948185312*b^3+1043522287029*b^2+1382133549472*b+ 96923129785)*(1+b)^2/(b-1)^14*n^2+104/98415*(4991119891*b^8+24470842088*b^7+ 38395615444*b^6+20114296664*b^5+2901251506*b^4+20114296664*b^3+38395615444*b^2+ 24470842088*b+4991119891)*(1+b)^3/(b-1)^14*n^3+64064/32805*(3631033*b^6+ 13221870*b^5+21628599*b^4+24102740*b^3+21628599*b^2+13221870*b+3631033)*(1+b)^4 /(b-1)^14*n^4+64064/729*(34555*b^4+96964*b^3+127410*b^2+96964*b+34555)*(1+b)^5/ (b-1)^14*n^5+448448/243*(217*b^2+380*b+217)*(1+b)^6/(b-1)^14*n^6+640640/81*(1+b )^7/(b-1)^14*n^7, 2/42687998325*(1+b)*(4862702300619701*b^14-5080572043802621*b ^13-92425168211792749*b^12-155700548061367546*b^11-852227219512047479*b^10-\ 1069602147282555955*b^9+3566619005983867695*b^8+7717811781203360340*b^7+ 3566619005983867695*b^6-1069602147282555955*b^5-852227219512047479*b^4-\ 155700548061367546*b^3-92425168211792749*b^2-5080572043802621*b+ 4862702300619701)/(b-1)^15-2/7971615*(4132897047839*b^12+39492064830840*b^11+ 77522711775126*b^10+665321944020056*b^9+2636078524261137*b^8+4270861143605808*b ^7+4498557028132980*b^6+4270861143605808*b^5+2636078524261137*b^4+ 665321944020056*b^3+77522711775126*b^2+39492064830840*b+4132897047839)*(1+b)^2/ (b-1)^15*n-8/295245*(33607281571*b^10-1175738402605*b^9-1902081492097*b^8+ 13703872585780*b^7+42716204657054*b^6+56149129286770*b^5+42716204657054*b^4+ 13703872585780*b^3-1902081492097*b^2-1175738402605*b+33607281571)*(1+b)^3/(b-1) ^15*n^2+104/885735*(231774050381*b^8+1433983575640*b^7+2217546807404*b^6+ 509244578920*b^5-807976013554*b^4+509244578920*b^3+2217546807404*b^2+ 1433983575640*b+231774050381)*(1+b)^4/(b-1)^15*n^3+64064/6561*(5140207*b^6+ 19216095*b^5+29007681*b^4+30769490*b^3+29007681*b^2+19216095*b+5140207)*(1+b)^5 /(b-1)^15*n^4+320320/243*(21577*b^4+54978*b^3+69394*b^2+54978*b+21577)*(1+b)^6/ (b-1)^15*n^5+22422400/729*(181*b^2+263*b+181)*(1+b)^7/(b-1)^15*n^6+22422400/81* (1+b)^8/(b-1)^15*n^7, 2/8537599665*(6449832315034223*b^16+25488004094654753*b^ 15-118764817851305310*b^14-429906530029624421*b^13-1389776469778337144*b^12-\ 4507515527051552919*b^11-1294539391368570562*b^10+18919668382786452443*b^9+ 33608870232999128658*b^8+18919668382786452443*b^7-1294539391368570562*b^6-\ 4507515527051552919*b^5-1389776469778337144*b^4-429906530029624421*b^3-\ 118764817851305310*b^2+25488004094654753*b+6449832315034223)/(b-1)^16-2/ 948622185*(1+b)*(987632706646077*b^14+38862356163131582*b^13+130388386703198079 *b^12+545177480837683692*b^11+3052643241610719509*b^10+8443564213287135138*b^9+ 12987244799027812143*b^8+14287208359294143336*b^7+12987244799027812143*b^6+ 8443564213287135138*b^5+3052643241610719509*b^4+545177480837683692*b^3+ 130388386703198079*b^2+38862356163131582*b+987632706646077)/(b-1)^16*n-4/ 7971615*(43810661897147*b^12-185355491962080*b^11-1610325377171346*b^10+ 2477696469665120*b^9+26251599534660405*b^8+61835010077554560*b^7+ 78875189721956676*b^6+61835010077554560*b^5+26251599534660405*b^4+ 2477696469665120*b^3-1610325377171346*b^2-185355491962080*b+43810661897147)*(1+ b)^2/(b-1)^16*n^2+16/885735*(7221322318051*b^10+79043527232254*b^9+ 242475276123207*b^8+267017058722856*b^7+8880691292886*b^6-174474639633996*b^5+ 8880691292886*b^4+267017058722856*b^3+242475276123207*b^2+79043527232254*b+ 7221322318051)*(1+b)^3/(b-1)^16*n^3+208/32805*(55199003297*b^8+341713351980*b^7 +833784795908*b^6+1177670803540*b^5+1262901725622*b^4+1177670803540*b^3+ 833784795908*b^2+341713351980*b+55199003297)*(1+b)^4/(b-1)^16*n^4+64064/6561*( 26419163*b^6+117761898*b^5+232185189*b^4+282058156*b^3+232185189*b^2+117761898* b+26419163)*(1+b)^5/(b-1)^16*n^5+640640/243*(26351*b^4+84168*b^3+116210*b^2+ 84168*b+26351)*(1+b)^6/(b-1)^16*n^6+2562560/81*(191*b^2+350*b+191)*(1+b)^7/(b-1 )^16*n^7+6406400/81*(1+b)^8/(b-1)^16*n^8] The limits of the scaled moments up to the , 16, -th are [0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025] This confirms that our random variable is asymptotically normal ---------------------------------- This ends this article that took, 2597.210, seconds to generate.