--------------------------------------- Explicit (Efficient!) Exact Expressions for the Factorial Moments of the Sum\ of a-Parking Functions up to the, 10, -th moment in terms of the length(n),a, and the Expectation Definition: An a-parking function of length n is a permutation of a weakly \ increasing sequence of positive integers x[1], ..., x[n] such that for all i x[i]<=a+i-1 For any a-parking function, p, let X[p] be its sum of entries Let E be the expectation. In this article we will present explicit expressi\ on in terms of E,n, and a, for all factorial moments up to the , 10, -th Theorem Number, 1, : the , 2, -th factorial moment of X is 2 3 2 3 2 2 3 n a 5 n n a (2/3 a + 2 n a + 1/3 a + n + 4/3 n - 1 + 1/3 a ) E - ----- - ---- - ---- 6 12 3 4 2 2 2 2 3 4 n a 3 2 n a n a n a 11 n a n n n - ---- - n a - ------ - ---- - --- - -------- - ---- - ---- - ---- 6 3 4 6 12 12 4 4 and in Maple format (2/3*a^2+2*n*a+1/3*a+n^2+4/3*n-1+1/3*a^3)*E-1/6*n^2*a^3-5/12*n^2-1/3*n*a^3-1/6* n*a^4-n^3*a-2/3*n^2*a-1/4*n*a^2-1/6*n*a-11/12*n^2*a^2-1/12*n-1/4*n^3-1/4*n^4 and in latex \left( 2/3\,{a}^{2}+2\,na+1/3\,a+{n}^{2}+4/3\,n-1+1/3\,{a}^{3} \right) E-1/6\,{n}^{2}{a}^{3}-{\frac {5}{12}}\,{n}^{2}-1/3\,n{a}^{3}-1 /6\,n{a}^{4}-{n}^{3}a-2/3\,{n}^{2}a-1/4\,n{a}^{2}-1/6\,na-{\frac {11}{ 12}}\,{n}^{2}{a}^{2}-1/12\,n-1/4\,{n}^{3}-1/4\,{n}^{4} Theorem Number, 2, : the , 3, -th factorial moment of X is / 6 19 5 4 79 3 2 19 4 3 83 2 131 |5/96 a + -- a + 3/4 n + -- n - 2 a - a + -- a - 1/2 a - -- n - --- n a \ 48 32 24 48 24 87 2 3 145 2 143 2 2 79 3 2 3 35 4 + -- n a + 3 n a + --- n a + --- n a + -- n a + 1/2 n a + -- n a 16 48 32 24 32 5 6 2 3 3 4 3 \ 39 n n 3 n a 27 n a 5 n a 217 n a + 2 - 17/4 n| E - ----- - ---- - ------- + ------- + ------ + -------- / 64 4 2 32 48 192 2 2 2 2 6 2 6 7 163 n a n a 27 n a 35 n a 19 n a 5 n a 5 n a + -------- + --- + -------- + ------- - ------- - ------- - ------ 96 2 16 48 96 192 192 3 3 4 3 2 4 3 4 5 2 5 4 207 n a n a 91 n a 51 n a 27 n a 23 n a 43 n a - --------- - ----- - -------- - -------- - ------- - -------- - ------- 64 4 48 64 64 32 16 5 3 2 4 2 2 3 4 3 n a 85 n a 199 n a 5 n 11 n n - ------ - -------- - --------- + ---- + ----- + ---- + n/4 2 24 64 4 48 96 and in Maple format (5/96*a^6+19/48*a^5+3/4*n^4+79/32*n^3-2*a^2-a+19/24*a^4-1/2*a^3-83/48*n^2-131/ 24*n*a+87/16*n^2*a+3*n^3*a+145/48*n*a^2+143/32*n^2*a^2+79/24*n*a^3+1/2*n^2*a^3+ 35/32*n*a^4+2-17/4*n)*E-39/64*n^5-1/4*n^6-3/2*n^2*a^3+27/32*n*a^3+5/48*n*a^4+ 217/192*n^3*a+163/96*n^2*a+1/2*n*a+27/16*n^2*a^2+35/48*n*a^2-19/96*n*a^6-5/192* n^2*a^6-5/192*n*a^7-207/64*n^3*a^3-1/4*n^4*a^3-91/48*n^2*a^4-51/64*n^3*a^4-27/ 64*n*a^5-23/32*n^2*a^5-43/16*n^4*a-3/2*n^5*a-85/24*n^3*a^2-199/64*n^4*a^2+5/4*n ^2+11/48*n^3+1/96*n^4+1/4*n and in latex \left( {\frac {5}{96}}\,{a}^{6}+{\frac {19}{48}}\,{a}^{5}+3/4\,{n}^{4} +{\frac {79}{32}}\,{n}^{3}-2\,{a}^{2}-a+{\frac {19}{24}}\,{a}^{4}-1/2\, {a}^{3}-{\frac {83}{48}}\,{n}^{2}-{\frac {131}{24}}\,na+{\frac {87}{16} }\,{n}^{2}a+3\,{n}^{3}a+{\frac {145}{48}}\,n{a}^{2}+{\frac {143}{32}}\, {n}^{2}{a}^{2}+{\frac {79}{24}}\,n{a}^{3}+1/2\,{n}^{2}{a}^{3}+{\frac { 35}{32}}\,n{a}^{4}+2-{\frac {17}{4}}\,n \right) E-{\frac {39}{64}}\,{n} ^{5}-1/4\,{n}^{6}-3/2\,{n}^{2}{a}^{3}+{\frac {27}{32}}\,n{a}^{3}+{ \frac {5}{48}}\,n{a}^{4}+{\frac {217}{192}}\,{n}^{3}a+{\frac {163}{96}} \,{n}^{2}a+1/2\,na+{\frac {27}{16}}\,{n}^{2}{a}^{2}+{\frac {35}{48}}\,n {a}^{2}-{\frac {19}{96}}\,n{a}^{6}-{\frac {5}{192}}\,{n}^{2}{a}^{6}-{ \frac {5}{192}}\,n{a}^{7}-{\frac {207}{64}}\,{n}^{3}{a}^{3}-1/4\,{n}^{4 }{a}^{3}-{\frac {91}{48}}\,{n}^{2}{a}^{4}-{\frac {51}{64}}\,{n}^{3}{a}^ {4}-{\frac {27}{64}}\,n{a}^{5}-{\frac {23}{32}}\,{n}^{2}{a}^{5}-{\frac {43}{16}}\,{n}^{4}a-3/2\,{n}^{5}a-{\frac {85}{24}}\,{n}^{3}{a}^{2}-{ \frac {199}{64}}\,{n}^{4}{a}^{2}+5/4\,{n}^{2}+{\frac {11}{48}}\,{n}^{3} +{\frac {1}{96}}\,{n}^{4}+1/4\,n Theorem Number, 3, : the , 4, -th factorial moment of X is / 11 8 79 9 331 7 281 4 457 6 199 5 47 5 6 |-6 + --- a + ----- a + --- a - --- a + --- a - --- a + -- n + 1/2 n \ 126 15120 720 60 720 120 16 3 2 1781 2 3 1327 3 697 4 433 3 + 3/5 a + 22/3 a + ---- n a - ---- n a - --- n a - --- n a 144 72 720 48 599 2 2603 2963 2 2 15683 2 151 6 2 6 - --- n a + ---- n a - ---- n a - ----- n a + --- n a + 5/48 n a 18 140 144 840 72 79 7 613 3 3 4 3 2 4 35 3 4 4063 5 + --- n a + --- n a + 1/2 n a + 49/4 n a + -- n a + ---- n a 336 48 16 720 161 2 5 4 5 3 2 127 4 2 6833 2 + --- n a + 49/4 n a + 3 n a + 59/3 n a + --- n a + ---- n 48 16 2520 6 7 6 2 6199 3 17 4 \ 85 n a 3 n a 155 n a - ---- n - -- n + 11/3 a + 97/6 n| E - ------- - ------ - --------- 432 24 / 16 2 32 5 2 6 3 5 3 10 2 8 8 12557 n a n a 145 n a 79 n a 1199 n a 703 n a - ----------- - ----- - --------- - -------- - ---------- - -------- 1008 4 16 30240 7560 3024 2 9 9 3 5 4 5 3 6 4 6 79 n a 11 n a 17729 n a 87 n a 78949 n a 5 n a - -------- - ------- - ----------- - -------- - ----------- - ------- 30240 252 2160 32 30240 96 3 7 2 7 5 6 2 3 3 19 n a 5357 n a 74917 n 121 n 8179 n a 1509 n a - -------- - ---------- + -------- + ------ + ---------- - --------- 112 4320 30240 504 840 560 4 3 2 2 2 2 2573 n a 9851 n a 3923 n a 11 n a 3539 n a 2209 n a + --------- + --------- - --------- - ------ - ---------- - --------- 5040 30240 720 6 1008 840 6 2 6 7 3 3 4 3 6763 n a 13373 n a 1537 n a 396269 n a 82153 n a + --------- - ----------- - --------- + ------------ - ----------- 10080 4320 4320 30240 6048 2 4 3 4 5 2 5 4 28309 n a 34115 n a 2559 n a 8759 n a 33419 n a + ----------- - ----------- + --------- + ---------- + ---------- 3024 6048 1120 15120 2520 5 3 2 4 2 2 3 4 3299 n a 582689 n a 58799 n a 73 n 731 n 10399 n + --------- + ------------ + ----------- - ----- + ------ + -------- 1008 30240 6048 16 1260 7560 7 8 5 4 4 4 27 n 3 n 109 n 43 n a 7909 n a - ----- - ---- - ----- - -------- - ---------- 32 16 120 32 720 and in Maple format (-6+11/126*a^8+79/15120*a^9+331/720*a^7-281/60*a^4+457/720*a^6-199/120*a^5+47/ 16*n^5+1/2*n^6+3/5*a^3+22/3*a^2+1781/144*n^2*a^3-1327/72*n*a^3-697/720*n*a^4-\ 433/48*n^3*a-599/18*n^2*a+2603/140*n*a-2963/144*n^2*a^2-15683/840*n*a^2+151/72* n*a^6+5/48*n^2*a^6+79/336*n*a^7+613/48*n^3*a^3+1/2*n^4*a^3+49/4*n^2*a^4+35/16*n ^3*a^4+4063/720*n*a^5+161/48*n^2*a^5+49/4*n^4*a+3*n^5*a+59/3*n^3*a^2+127/16*n^4 *a^2+6833/2520*n^2-6199/432*n^3-17/24*n^4+11/3*a+97/6*n)*E-85/16*n^6*a-3/2*n^7* a-155/32*n^6*a^2-12557/1008*n^5*a^2-1/4*n^6*a^3-145/16*n^5*a^3-79/30240*n*a^10-\ 1199/7560*n^2*a^8-703/3024*n*a^8-79/30240*n^2*a^9-11/252*n*a^9-17729/2160*n^3*a ^5-87/32*n^4*a^5-78949/30240*n^3*a^6-5/96*n^4*a^6-19/112*n^3*a^7-5357/4320*n^2* a^7+74917/30240*n^5+121/504*n^6+8179/840*n^2*a^3-1509/560*n*a^3+2573/5040*n*a^4 +9851/30240*n^3*a-3923/720*n^2*a-11/6*n*a-3539/1008*n^2*a^2-2209/840*n*a^2+6763 /10080*n*a^6-13373/4320*n^2*a^6-1537/4320*n*a^7+396269/30240*n^3*a^3-82153/6048 *n^4*a^3+28309/3024*n^2*a^4-34115/6048*n^3*a^4+2559/1120*n*a^5+8759/15120*n^2*a ^5+33419/2520*n^4*a+3299/1008*n^5*a+582689/30240*n^3*a^2+58799/6048*n^4*a^2-73/ 16*n^2+731/1260*n^3+10399/7560*n^4-27/32*n^7-3/16*n^8-109/120*n-43/32*n^5*a^4-\ 7909/720*n^4*a^4 and in latex \left( -6+{\frac {11}{126}}\,{a}^{8}+{\frac {79}{15120}}\,{a}^{9}+{ \frac {331}{720}}\,{a}^{7}-{\frac {281}{60}}\,{a}^{4}+{\frac {457}{720} }\,{a}^{6}-{\frac {199}{120}}\,{a}^{5}+{\frac {47}{16}}\,{n}^{5}+1/2\,{ n}^{6}+3/5\,{a}^{3}+{\frac {22}{3}}\,{a}^{2}+{\frac {1781}{144}}\,{n}^{ 2}{a}^{3}-{\frac {1327}{72}}\,n{a}^{3}-{\frac {697}{720}}\,n{a}^{4}-{ \frac {433}{48}}\,{n}^{3}a-{\frac {599}{18}}\,{n}^{2}a+{\frac {2603}{ 140}}\,na-{\frac {2963}{144}}\,{n}^{2}{a}^{2}-{\frac {15683}{840}}\,n{a }^{2}+{\frac {151}{72}}\,n{a}^{6}+{\frac {5}{48}}\,{n}^{2}{a}^{6}+{ \frac {79}{336}}\,n{a}^{7}+{\frac {613}{48}}\,{n}^{3}{a}^{3}+1/2\,{n}^{ 4}{a}^{3}+{\frac {49}{4}}\,{n}^{2}{a}^{4}+{\frac {35}{16}}\,{n}^{3}{a}^ {4}+{\frac {4063}{720}}\,n{a}^{5}+{\frac {161}{48}}\,{n}^{2}{a}^{5}+{ \frac {49}{4}}\,{n}^{4}a+3\,{n}^{5}a+{\frac {59}{3}}\,{n}^{3}{a}^{2}+{ \frac {127}{16}}\,{n}^{4}{a}^{2}+{\frac {6833}{2520}}\,{n}^{2}-{\frac { 6199}{432}}\,{n}^{3}-{\frac {17}{24}}\,{n}^{4}+11/3\,a+{\frac {97}{6}} \,n \right) E-{\frac {85}{16}}\,{n}^{6}a-3/2\,{n}^{7}a-{\frac {155}{32} }\,{n}^{6}{a}^{2}-{\frac {12557}{1008}}\,{n}^{5}{a}^{2}-1/4\,{n}^{6}{a} ^{3}-{\frac {145}{16}}\,{n}^{5}{a}^{3}-{\frac {79}{30240}}\,n{a}^{10}-{ \frac {1199}{7560}}\,{n}^{2}{a}^{8}-{\frac {703}{3024}}\,n{a}^{8}-{ \frac {79}{30240}}\,{n}^{2}{a}^{9}-{\frac {11}{252}}\,n{a}^{9}-{\frac { 17729}{2160}}\,{n}^{3}{a}^{5}-{\frac {87}{32}}\,{n}^{4}{a}^{5}-{\frac { 78949}{30240}}\,{n}^{3}{a}^{6}-{\frac {5}{96}}\,{n}^{4}{a}^{6}-{\frac { 19}{112}}\,{n}^{3}{a}^{7}-{\frac {5357}{4320}}\,{n}^{2}{a}^{7}+{\frac { 74917}{30240}}\,{n}^{5}+{\frac {121}{504}}\,{n}^{6}+{\frac {8179}{840}} \,{n}^{2}{a}^{3}-{\frac {1509}{560}}\,n{a}^{3}+{\frac {2573}{5040}}\,n{ a}^{4}+{\frac {9851}{30240}}\,{n}^{3}a-{\frac {3923}{720}}\,{n}^{2}a-{ \frac {11}{6}}\,na-{\frac {3539}{1008}}\,{n}^{2}{a}^{2}-{\frac {2209}{ 840}}\,n{a}^{2}+{\frac {6763}{10080}}\,n{a}^{6}-{\frac {13373}{4320}}\, {n}^{2}{a}^{6}-{\frac {1537}{4320}}\,n{a}^{7}+{\frac {396269}{30240}}\, {n}^{3}{a}^{3}-{\frac {82153}{6048}}\,{n}^{4}{a}^{3}+{\frac {28309}{ 3024}}\,{n}^{2}{a}^{4}-{\frac {34115}{6048}}\,{n}^{3}{a}^{4}+{\frac { 2559}{1120}}\,n{a}^{5}+{\frac {8759}{15120}}\,{n}^{2}{a}^{5}+{\frac { 33419}{2520}}\,{n}^{4}a+{\frac {3299}{1008}}\,{n}^{5}a+{\frac {582689}{ 30240}}\,{n}^{3}{a}^{2}+{\frac {58799}{6048}}\,{n}^{4}{a}^{2}-{\frac { 73}{16}}\,{n}^{2}+{\frac {731}{1260}}\,{n}^{3}+{\frac {10399}{7560}}\,{ n}^{4}-{\frac {27}{32}}\,{n}^{7}-3/16\,{n}^{8}-{\frac {109}{120}}\,n-{ \frac {43}{32}}\,{n}^{5}{a}^{4}-{\frac {7909}{720}}\,{n}^{4}{a}^{4} Theorem Number, 4, : the , 5, -th factorial moment of X is / 167159 10 3257 11 445 12 6017 8 12751 9 1555 7 |24 + ------- a + ------ a + ------- a + ----- a + ----- a - ---- a \ 1451520 290304 1161216 24192 26880 432 1675 6 7 1825 6 2 143935 5 2 6 3 + ---- n a + 5/2 n a + ---- n a + ------ n a + 5/12 n a 96 192 3072 2275 5 3 5785 10 117365 2 8 647821 8 79 2 9 + ---- n a + ------ n a + ------ n a + ------ n a + ---- n a 96 193536 129024 193536 6048 158911 9 99239 3 5 355 4 5 48745 3 6 25 4 6 + ------ n a + ----- n a + --- n a + ----- n a + --- n a 290304 2304 48 4608 192 395 3 7 44797 2 7 649 4 3601 6 313 5 710365 5 + --- n a + ----- n a + --- a - ---- a + --- a - ------ n 672 5376 24 480 48 27648 9695 6 3 2 1117045 2 3 204443 3 2309333 4 + ---- n + 3/2 a - 100/3 a - ------- n a + ------ n a - ------- n a 6144 8064 2016 120960 769555 3 2375579 2 133501 2201225 2 2 24903 2 + ------ n a + ------- n a - ------ n a + ------- n a + ----- n a 72576 12096 1680 24192 224 13141 6 88991 2 6 153319 7 903785 3 3 309355 4 3 - ----- n a + ----- n a + ------ n a - ------ n a + ------ n a 1152 3456 24192 10368 4608 1463999 2 4 207913 3 4 3234773 5 877 2 5 793805 4 - ------- n a + ------ n a - ------- n a + --- n a - ------ n a 13824 4608 60480 432 6912 795 5 28129361 3 2 142885 4 2 24587 2 433955 3 - --- n a - -------- n a - ------ n a + ----- n + ------ n 512 145152 4608 20160 5376 3815321 4 545 7 8 175 5 4 - ------- n + --- n + 5/16 n - 50/3 a - 603/8 n + --- n a 290304 192 64 13 11 2 11 12 2 246815 4 4\ 445 a n 47293 a n 670861 a n 445 a n + ------ n a | E - --------- - ------------ - ------------ - ---------- 6144 / 2322432 2322432 11612160 2322432 12 2 10 3 10 4 8 4 9 3257 a n 1893083 n a 2771 n a 574559 n a 79 n a - ---------- - -------------- - ----------- - ------------ - -------- 580608 5806080 129024 774144 12096 3 9 5 7 3 8 6 7 25901 n a 965 n a 6510745 n a 11250691 n a 32855 n a - ----------- - --------- - ------------- + ------------- + ---------- 36288 2688 1161216 258048 14336 6 2 5 2 6 3 5 3 229045 n a 1562623 n a 11982265 n a 108178031 n a + ------------ + ------------- - -------------- + --------------- 14336 14336 258048 2322432 10 2 8 8 2 9 9 469151 n a 3272503 n a 1196551 n a 623099 n a 36959 n a - ------------ - ------------- + ------------ - ------------ - ---------- 1935360 967680 725760 331776 215040 3 5 4 5 3 6 4 6 12852101 n a 122121721 n a 2542711 n a 6105947 n a + -------------- - --------------- + ------------- - ------------- 193536 5806080 967680 193536 9 10 3 7 2 7 5 6 115 n n 29570231 n a 171281 n a 13347119 n 783365 n - ------ - --- - -------------- + ------------ - ----------- + --------- 128 8 1935360 27648 1451520 290304 5 5 5 6 6 6 4 7 6 4 152843 n a 1683709 n a 25 n a 1164851 n a 1050077 n a - ------------ - ------------- - -------- - ------------- - ------------- 4096 193536 384 129024 36864 7 4 8 9 7 2 8 2 605 n a 6 5 355 n a 5 n a 3254155 n a 2105 n a - --------- - 5 n a - -------- - ------ - ------------- - ---------- 384 48 4 129024 384 7 3 8 3 2 3 3 4 5755 n a 5 n a 28783 n a 72937 n a 104767 n a - ---------- - ------- - ----------- + ---------- - ----------- 384 24 504 6720 20160 3 2 2 2 2 7874459 n a 888337 n a 25 n a 202163 n a 39701 n a - ------------ + ----------- + ------ + ------------ + ---------- 483840 40320 3 30240 3360 6 2 6 7 3 3 9013 n a 13515221 n a 467777 n a 47497343 n a - --------- + -------------- + ----------- - -------------- 5040 483840 120960 967680 4 3 2 4 3 4 5 9444179 n a 1843777 n a 272567 n a 593165 n a + ------------- - ------------- + ------------ - ----------- 72576 40320 3780 48384 2 5 4 5 3 2 8392859 n a 588487 n a 3191381 n a 149475217 n a + ------------- - ----------- + ------------ - --------------- 725760 9216 1161216 1451520 4 2 2 3 4 7 8 14296861 n a 165 n 156887 n 793193 n 810983 n 21835 n - -------------- + ------ - --------- - --------- + --------- - -------- 725760 8 20160 80640 129024 258048 5 4 4 4 49 n 758257 n a 218086867 n a + ---- - ------------ + --------------- 12 16128 2903040 and in Maple format (24+167159/1451520*a^10+3257/290304*a^11+445/1161216*a^12+6017/24192*a^8+12751/ 26880*a^9-1555/432*a^7+1675/96*n^6*a+5/2*n^7*a+1825/192*n^6*a^2+143935/3072*n^5 *a^2+5/12*n^6*a^3+2275/96*n^5*a^3+5785/193536*n*a^10+117365/129024*n^2*a^8+ 647821/193536*n*a^8+79/6048*n^2*a^9+158911/290304*n*a^9+99239/2304*n^3*a^5+355/ 48*n^4*a^5+48745/4608*n^3*a^6+25/192*n^4*a^6+395/672*n^3*a^7+44797/5376*n^2*a^7 +649/24*a^4-3601/480*a^6+313/48*a^5-710365/27648*n^5+9695/6144*n^6+3/2*a^3-100/ 3*a^2-1117045/8064*n^2*a^3+204443/2016*n*a^3-2309333/120960*n*a^4+769555/72576* n^3*a+2375579/12096*n^2*a-133501/1680*n*a+2201225/24192*n^2*a^2+24903/224*n*a^2 -13141/1152*n*a^6+88991/3456*n^2*a^6+153319/24192*n*a^7-903785/10368*n^3*a^3+ 309355/4608*n^4*a^3-1463999/13824*n^2*a^4+207913/4608*n^3*a^4-3234773/60480*n*a ^5+877/432*n^2*a^5-793805/6912*n^4*a-795/512*n^5*a-28129361/145152*n^3*a^2-\ 142885/4608*n^4*a^2+24587/20160*n^2+433955/5376*n^3-3815321/290304*n^4+545/192* n^7+5/16*n^8-50/3*a-603/8*n+175/64*n^5*a^4+246815/6144*n^4*a^4)*E-445/2322432*a ^13*n-47293/2322432*a^11*n^2-670861/11612160*a^11*n-445/2322432*a^12*n^2-3257/ 580608*a^12*n-1893083/5806080*n^2*a^10-2771/129024*n^3*a^10-574559/774144*n^4*a ^8-79/12096*n^4*a^9-25901/36288*n^3*a^9-965/2688*n^5*a^7-6510745/1161216*n^3*a^ 8+11250691/258048*n^6*a+32855/14336*n^7*a+229045/14336*n^6*a^2+1562623/14336*n^ 5*a^2-11982265/258048*n^6*a^3+108178031/2322432*n^5*a^3-469151/1935360*n*a^10-\ 3272503/967680*n^2*a^8+1196551/725760*n*a^8-623099/331776*n^2*a^9-36959/215040* n*a^9+12852101/193536*n^3*a^5-122121721/5806080*n^4*a^5+2542711/967680*n^3*a^6-\ 6105947/193536*n^4*a^6-115/128*n^9-1/8*n^10-29570231/1935360*n^3*a^7+171281/ 27648*n^2*a^7-13347119/1451520*n^5+783365/290304*n^6-152843/4096*n^5*a^5-\ 1683709/193536*n^5*a^6-25/384*n^6*a^6-1164851/129024*n^4*a^7-1050077/36864*n^6* a^4-605/384*n^7*a^4-5*n^6*a^5-355/48*n^8*a-5/4*n^9*a-3254155/129024*n^7*a^2-\ 2105/384*n^8*a^2-5755/384*n^7*a^3-5/24*n^8*a^3-28783/504*n^2*a^3+72937/6720*n*a ^3-104767/20160*n*a^4-7874459/483840*n^3*a+888337/40320*n^2*a+25/3*n*a+202163/ 30240*n^2*a^2+39701/3360*n*a^2-9013/5040*n*a^6+13515221/483840*n^2*a^6+467777/ 120960*n*a^7-47497343/967680*n^3*a^3+9444179/72576*n^4*a^3-1843777/40320*n^2*a^ 4+272567/3780*n^3*a^4-593165/48384*n*a^5+8392859/725760*n^2*a^5-588487/9216*n^4 *a+3191381/1161216*n^5*a-149475217/1451520*n^3*a^2-14296861/725760*n^4*a^2+165/ 8*n^2-156887/20160*n^3-793193/80640*n^4+810983/129024*n^7-21835/258048*n^8+49/ 12*n-758257/16128*n^5*a^4+218086867/2903040*n^4*a^4 and in latex \left( 24+{\frac {167159}{1451520}}\,{a}^{10}+{\frac {3257}{290304}}\, {a}^{11}+{\frac {445}{1161216}}\,{a}^{12}+{\frac {6017}{24192}}\,{a}^{8 }+{\frac {12751}{26880}}\,{a}^{9}-{\frac {1555}{432}}\,{a}^{7}+{\frac { 1675}{96}}\,{n}^{6}a+5/2\,{n}^{7}a+{\frac {1825}{192}}\,{n}^{6}{a}^{2}+ {\frac {143935}{3072}}\,{n}^{5}{a}^{2}+{\frac {5}{12}}\,{n}^{6}{a}^{3}+ {\frac {2275}{96}}\,{n}^{5}{a}^{3}+{\frac {5785}{193536}}\,n{a}^{10}+{ \frac {117365}{129024}}\,{n}^{2}{a}^{8}+{\frac {647821}{193536}}\,n{a}^ {8}+{\frac {79}{6048}}\,{n}^{2}{a}^{9}+{\frac {158911}{290304}}\,n{a}^{ 9}+{\frac {99239}{2304}}\,{n}^{3}{a}^{5}+{\frac {355}{48}}\,{n}^{4}{a}^ {5}+{\frac {48745}{4608}}\,{n}^{3}{a}^{6}+{\frac {25}{192}}\,{n}^{4}{a} ^{6}+{\frac {395}{672}}\,{n}^{3}{a}^{7}+{\frac {44797}{5376}}\,{n}^{2}{ a}^{7}+{\frac {649}{24}}\,{a}^{4}-{\frac {3601}{480}}\,{a}^{6}+{\frac { 313}{48}}\,{a}^{5}-{\frac {710365}{27648}}\,{n}^{5}+{\frac {9695}{6144} }\,{n}^{6}+3/2\,{a}^{3}-{\frac {100}{3}}\,{a}^{2}-{\frac {1117045}{8064 }}\,{n}^{2}{a}^{3}+{\frac {204443}{2016}}\,n{a}^{3}-{\frac {2309333}{ 120960}}\,n{a}^{4}+{\frac {769555}{72576}}\,{n}^{3}a+{\frac {2375579}{ 12096}}\,{n}^{2}a-{\frac {133501}{1680}}\,na+{\frac {2201225}{24192}}\, {n}^{2}{a}^{2}+{\frac {24903}{224}}\,n{a}^{2}-{\frac {13141}{1152}}\,n{ a}^{6}+{\frac {88991}{3456}}\,{n}^{2}{a}^{6}+{\frac {153319}{24192}}\,n {a}^{7}-{\frac {903785}{10368}}\,{n}^{3}{a}^{3}+{\frac {309355}{4608}} \,{n}^{4}{a}^{3}-{\frac {1463999}{13824}}\,{n}^{2}{a}^{4}+{\frac { 207913}{4608}}\,{n}^{3}{a}^{4}-{\frac {3234773}{60480}}\,n{a}^{5}+{ \frac {877}{432}}\,{n}^{2}{a}^{5}-{\frac {793805}{6912}}\,{n}^{4}a-{ \frac {795}{512}}\,{n}^{5}a-{\frac {28129361}{145152}}\,{n}^{3}{a}^{2}- {\frac {142885}{4608}}\,{n}^{4}{a}^{2}+{\frac {24587}{20160}}\,{n}^{2}+ {\frac {433955}{5376}}\,{n}^{3}-{\frac {3815321}{290304}}\,{n}^{4}+{ \frac {545}{192}}\,{n}^{7}+{\frac {5}{16}}\,{n}^{8}-{\frac {50}{3}}\,a- {\frac {603}{8}}\,n+{\frac {175}{64}}\,{n}^{5}{a}^{4}+{\frac {246815}{ 6144}}\,{n}^{4}{a}^{4} \right) E-{\frac {445}{2322432}}\,{a}^{13}n-{ \frac {47293}{2322432}}\,{a}^{11}{n}^{2}-{\frac {670861}{11612160}}\,{a }^{11}n-{\frac {445}{2322432}}\,{a}^{12}{n}^{2}-{\frac {3257}{580608}} \,{a}^{12}n-{\frac {1893083}{5806080}}\,{n}^{2}{a}^{10}-{\frac {2771}{ 129024}}\,{n}^{3}{a}^{10}-{\frac {574559}{774144}}\,{n}^{4}{a}^{8}-{ \frac {79}{12096}}\,{n}^{4}{a}^{9}-{\frac {25901}{36288}}\,{n}^{3}{a}^{ 9}-{\frac {965}{2688}}\,{n}^{5}{a}^{7}-{\frac {6510745}{1161216}}\,{n}^ {3}{a}^{8}+{\frac {11250691}{258048}}\,{n}^{6}a+{\frac {32855}{14336}} \,{n}^{7}a+{\frac {229045}{14336}}\,{n}^{6}{a}^{2}+{\frac {1562623}{ 14336}}\,{n}^{5}{a}^{2}-{\frac {11982265}{258048}}\,{n}^{6}{a}^{3}+{ \frac {108178031}{2322432}}\,{n}^{5}{a}^{3}-{\frac {469151}{1935360}}\, n{a}^{10}-{\frac {3272503}{967680}}\,{n}^{2}{a}^{8}+{\frac {1196551}{ 725760}}\,n{a}^{8}-{\frac {623099}{331776}}\,{n}^{2}{a}^{9}-{\frac { 36959}{215040}}\,n{a}^{9}+{\frac {12852101}{193536}}\,{n}^{3}{a}^{5}-{ \frac {122121721}{5806080}}\,{n}^{4}{a}^{5}+{\frac {2542711}{967680}}\, {n}^{3}{a}^{6}-{\frac {6105947}{193536}}\,{n}^{4}{a}^{6}-{\frac {115}{ 128}}\,{n}^{9}-1/8\,{n}^{10}-{\frac {29570231}{1935360}}\,{n}^{3}{a}^{7 }+{\frac {171281}{27648}}\,{n}^{2}{a}^{7}-{\frac {13347119}{1451520}}\, {n}^{5}+{\frac {783365}{290304}}\,{n}^{6}-{\frac {152843}{4096}}\,{n}^{ 5}{a}^{5}-{\frac {1683709}{193536}}\,{n}^{5}{a}^{6}-{\frac {25}{384}}\, {n}^{6}{a}^{6}-{\frac {1164851}{129024}}\,{n}^{4}{a}^{7}-{\frac { 1050077}{36864}}\,{n}^{6}{a}^{4}-{\frac {605}{384}}\,{n}^{7}{a}^{4}-5\, {n}^{6}{a}^{5}-{\frac {355}{48}}\,{n}^{8}a-5/4\,{n}^{9}a-{\frac { 3254155}{129024}}\,{n}^{7}{a}^{2}-{\frac {2105}{384}}\,{n}^{8}{a}^{2}-{ \frac {5755}{384}}\,{n}^{7}{a}^{3}-{\frac {5}{24}}\,{n}^{8}{a}^{3}-{ \frac {28783}{504}}\,{n}^{2}{a}^{3}+{\frac {72937}{6720}}\,n{a}^{3}-{ \frac {104767}{20160}}\,n{a}^{4}-{\frac {7874459}{483840}}\,{n}^{3}a+{ \frac {888337}{40320}}\,{n}^{2}a+{\frac {25}{3}}\,na+{\frac {202163}{ 30240}}\,{n}^{2}{a}^{2}+{\frac {39701}{3360}}\,n{a}^{2}-{\frac {9013}{ 5040}}\,n{a}^{6}+{\frac {13515221}{483840}}\,{n}^{2}{a}^{6}+{\frac { 467777}{120960}}\,n{a}^{7}-{\frac {47497343}{967680}}\,{n}^{3}{a}^{3}+{ \frac {9444179}{72576}}\,{n}^{4}{a}^{3}-{\frac {1843777}{40320}}\,{n}^{ 2}{a}^{4}+{\frac {272567}{3780}}\,{n}^{3}{a}^{4}-{\frac {593165}{48384} }\,n{a}^{5}+{\frac {8392859}{725760}}\,{n}^{2}{a}^{5}-{\frac {588487}{ 9216}}\,{n}^{4}a+{\frac {3191381}{1161216}}\,{n}^{5}a-{\frac {149475217 }{1451520}}\,{n}^{3}{a}^{2}-{\frac {14296861}{725760}}\,{n}^{4}{a}^{2}+ {\frac {165}{8}}\,{n}^{2}-{\frac {156887}{20160}}\,{n}^{3}-{\frac { 793193}{80640}}\,{n}^{4}+{\frac {810983}{129024}}\,{n}^{7}-{\frac { 21835}{258048}}\,{n}^{8}+{\frac {49}{12}}\,n-{\frac {758257}{16128}}\,{ n}^{5}{a}^{4}+{\frac {218086867}{2903040}}\,{n}^{4}{a}^{4} Theorem Number, 5, : the , 6, -th factorial moment of X is / 18323 15 82949 14 596971 10 309959 11 8527271 12 |-120 + --------- a + -------- a - ------- a + ------ a + -------- a \ 830269440 83026944 1451520 725760 63866880 18323 13 592717 11 2 7406857 11 445 12 2 + ------- a n + ------- a n + ------- a n + ------ a n 6918912 4257792 7983360 387072 263399 12 264877 2 10 5785 3 10 368521 8 1482059 9 + ------- a n + ------ n a + ----- n a - ------ a - ------- a 3193344 107520 64512 40320 241920 29969 7 1083331 13 14755 4 8 79 4 9 645017 3 9 + ----- a + -------- a + ----- n a + ---- n a + ------ n a 1260 63866880 6144 4032 193536 395 5 7 1955791 3 8 4059055 6 52775 7 22325 6 2 + --- n a + ------- n a - ------- n a + ----- n a + ----- n a 448 64512 18432 3072 3072 123102107 5 2 1024415 6 3 848665 5 3 12776047 10 - --------- n a + ------- n a - ------ n a + -------- n a 193536 6144 6912 2903040 2013193 2 8 4838689 8 13454419 2 9 1799351 9 + ------- n a - ------- n a + -------- n a + ------- n a 55296 138240 829440 414720 760032671 3 5 3013205 4 5 892613 3 6 1351567 4 6 155 9 - --------- n a + ------- n a + ------ n a + ------- n a + --- n 1451520 18432 41472 9216 64 10 204077 3 7 5203039 2 7 4139 4 134143 6 2867 5 + 3/16 n + ------ n a - ------- n a - ---- a + ------ a - ---- a 2016 96768 24 2016 112 110328067 5 21739835 6 199841 5 5 36665 5 6 25 6 6 + --------- n - -------- n + ------ n a + ----- n a + --- n a 580608 387072 1536 1536 192 1455469 4 7 470365 6 4 175 7 4 2035 6 5 315 8 + ------- n a + ------ n a + --- n a + ---- n a + --- n a 43008 6144 64 192 16 9 228445 7 2 595 8 2 6035 7 3 8 3 + 15/8 n a + ------ n a + --- n a + ---- n a + 5/16 n a 3072 64 192 3 2 24743447 2 3 11444669 3 111337351 4 - 161/6 a + 548/3 a + -------- n a - -------- n a + --------- n a 20160 18480 443520 16328653 3 35256667 2 59493173 27187643 2 2 + -------- n a - -------- n a + -------- n a - -------- n a 103680 27720 144144 63360 1046262989 2 426487 6 85871789 2 6 12666721 7 - ---------- n a + ------ n a - -------- n a - -------- n a 1441440 13440 241920 120960 16413223 3 3 288303509 4 3 377926007 2 4 1178697433 3 4 + -------- n a - --------- n a + --------- n a - ---------- n a 36288 290304 483840 1451520 17483839 5 123525077 2 5 265617553 4 1657177 5 + -------- n a - --------- n a + --------- n a - ------- n a 40320 483840 290304 9216 33600853 3 2 21688951 4 2 40215727 2 1329871843 3 + -------- n a - -------- n a - -------- n - ---------- n 20736 290304 576576 2661120 253709107 4 302195 7 24605 8 10103 28987 5 4 + --------- n - ------ n + ----- n + 274/3 a + ----- n + ----- n a 1451520 9216 6144 24 128 13 11 2 11 21223417 4 4\ 954251 a n 189748607 a n 23112937 a n - -------- n a | E - ------------ - ---------------- + -------------- 55296 / 14192640 79073280 150958080 12 2 12 2 10 3 10 66985679 a n 367127357 a n 112096561 n a 1037207609 n a - --------------- - --------------- - ---------------- - ----------------- 127733760 1660538880 50319360 103783680 11 12 4 8 4 9 3 9 105 n 5 n 463556473 n a 11994083 n a 753811993 n a - ------- - ----- - --------------- - -------------- - --------------- 128 64 7188480 539136 37739520 5 7 3 8 16 15 2 1429555837 n a 7943164001 n a 18323 a n 18323 a n - ---------------- + ---------------- - ----------- - ------------ 12579840 207567360 1660538880 1660538880 15 6 7 6 2 82949 a n 506426379131 n a 975591389 n a 102632415811 n a - ----------- - ----------------- + -------------- + ------------------ 166053888 1660538880 18450432 1660538880 5 2 6 3 5 3 4835193533 n a 9891176045 n a 189808205033 n a - ---------------- + ---------------- - ------------------ 5930496 15814656 1660538880 10 2 8 8 2 9 244337581 n a 760244515 n a 999785161 n a 6215424169 n a + --------------- + --------------- - -------------- + ---------------- 83026944 13837824 103783680 332107776 9 3 5 4 5 3 6 91397377 n a 63421301017 n a 29299394119 n a 125169465 n a + ------------- - ----------------- + ----------------- + --------------- 18450432 138378240 63866880 1025024 4 6 9 10 3 7 8957262229 n a 60327947 n 21145 n 6362511781 n a + ---------------- + ----------- - --------- + ---------------- 23721984 6150144 28672 30750720 2 7 5 6 5 5 606079027 n a 6678891823 n 6676860247 n 210564222299 n a - --------------- + ------------- - ------------- + ------------------ 46126080 207567360 207567360 553512960 5 6 6 6 4 7 65179143199 n a 1261639391 n a 10104223247 n a - ----------------- - ---------------- + ----------------- 830269440 10063872 830269440 6 4 7 4 6 5 8 10962997099 n a 1113683 n a 89672259161 n a 1611453643 n a + ----------------- - ------------- - ----------------- + --------------- 50319360 6864 553512960 18450432 9 7 2 8 2 7 3 11625 n a 364883215 n a 34052075 n a 7107800201 n a - ---------- + --------------- + -------------- + ---------------- 3584 1153152 9225216 110702592 8 3 13 3 13 2 2 3 12721165 n a 70073 a n 6290323 a n 524453921 n a - -------------- - ------------ - -------------- + --------------- 129024 36900864 127733760 1441440 3 4 3 2 51177667 n a 14618537 n a 809534527 n a 1263751 n a 137 n a - ------------- + ------------- + -------------- - ------------ - ------- 960960 360360 5322240 11648 3 2 2 2 6 2 6 776983 n a 18502223 n a 804865 n a 2965016987 n a + ------------ - ------------- + ----------- - ---------------- 174720 288288 768768 13837824 7 3 3 4 3 561198893 n a 24377923049 n a 215204446537 n a - -------------- + ----------------- - ------------------ 17297280 138378240 207567360 2 4 3 4 5 2 5 857863687 n a 2680206731 n a 2536567109 n a 15126824513 n a + --------------- - ---------------- + --------------- - ----------------- 3459456 4193280 34594560 103783680 4 5 3 2 140172576161 n a 908847761 n a 63379104641 n a + ----------------- - -------------- + ----------------- 415134720 7907328 103783680 4 2 12 4 12 3 2 3 22256344267 n a 445 a n 30313853 a n 3597 n 529203 n - ----------------- - ---------- - --------------- - ------- + --------- 415134720 774144 276756480 32 8008 4 11 4 11 3 7 2241295039 n 30301 a n 211788041 a n 32725526759 n + ------------- - ------------ - ---------------- - -------------- 34594560 266112 127733760 830269440 8 10 5 10 4 9 6 9 5 170554903 n 1007 a n 12053785 a n 79 a n 1092667 a n + ------------ - ----------- - --------------- - -------- - ------------- 15814656 18432 4257792 8064 387072 6 8 5 8 6 7 7 7 8 6 421087 n a 14651813 n a 1021799 n a 85 n a 25 n a - ------------ - -------------- - ------------- - -------- - -------- 258048 479232 36864 168 384 7 6 8 5 7 5 9 4 9 3 1084037 n a 845 n a 3437389 n a 195 n a 75 n a - ------------- - --------- - ------------- - --------- - -------- 64512 128 36864 128 4 10 3 8 4 9 2 10 2 5 n a 602141 n a 538585 n a 665 n a 11153 n - -------- - ------------ - ------------ - ---------- - ------- 32 12288 14336 128 504 5 4 4 4 14 2 14 1197321563507 n a 9417813589 n a 1003309 a n 30523 a n + ------------------- - ---------------- - -------------- - ----------- 1660538880 23063040 553512960 3594240 11 10 15 n a 535 n a - -------- - --------- 16 64 and in Maple format (-120+18323/830269440*a^15+82949/83026944*a^14-596971/1451520*a^10+309959/ 725760*a^11+8527271/63866880*a^12+18323/6918912*a^13*n+592717/4257792*a^11*n^2+ 7406857/7983360*a^11*n+445/387072*a^12*n^2+263399/3193344*a^12*n+264877/107520* n^2*a^10+5785/64512*n^3*a^10-368521/40320*a^8-1482059/241920*a^9+29969/1260*a^7 +1083331/63866880*a^13+14755/6144*n^4*a^8+79/4032*n^4*a^9+645017/193536*n^3*a^9 +395/448*n^5*a^7+1955791/64512*n^3*a^8-4059055/18432*n^6*a+52775/3072*n^7*a+ 22325/3072*n^6*a^2-123102107/193536*n^5*a^2+1024415/6144*n^6*a^3-848665/6912*n^ 5*a^3+12776047/2903040*n*a^10+2013193/55296*n^2*a^8-4838689/138240*n*a^8+ 13454419/829440*n^2*a^9+1799351/414720*n*a^9-760032671/1451520*n^3*a^5+3013205/ 18432*n^4*a^5+892613/41472*n^3*a^6+1351567/9216*n^4*a^6+155/64*n^9+3/16*n^10+ 204077/2016*n^3*a^7-5203039/96768*n^2*a^7-4139/24*a^4+134143/2016*a^6-2867/112* a^5+110328067/580608*n^5-21739835/387072*n^6+199841/1536*n^5*a^5+36665/1536*n^5 *a^6+25/192*n^6*a^6+1455469/43008*n^4*a^7+470365/6144*n^6*a^4+175/64*n^7*a^4+ 2035/192*n^6*a^5+315/16*n^8*a+15/8*n^9*a+228445/3072*n^7*a^2+595/64*n^8*a^2+ 6035/192*n^7*a^3+5/16*n^8*a^3-161/6*a^3+548/3*a^2+24743447/20160*n^2*a^3-\ 11444669/18480*n*a^3+111337351/443520*n*a^4+16328653/103680*n^3*a-35256667/ 27720*n^2*a+59493173/144144*n*a-27187643/63360*n^2*a^2-1046262989/1441440*n*a^2 +426487/13440*n*a^6-85871789/241920*n^2*a^6-12666721/120960*n*a^7+16413223/ 36288*n^3*a^3-288303509/290304*n^4*a^3+377926007/483840*n^2*a^4-1178697433/ 1451520*n^3*a^4+17483839/40320*n*a^5-123525077/483840*n^2*a^5+265617553/290304* n^4*a-1657177/9216*n^5*a+33600853/20736*n^3*a^2-21688951/290304*n^4*a^2-\ 40215727/576576*n^2-1329871843/2661120*n^3+253709107/1451520*n^4-302195/9216*n^ 7+24605/6144*n^8+274/3*a+10103/24*n+28987/128*n^5*a^4-21223417/55296*n^4*a^4)*E -954251/14192640*a^13*n-189748607/79073280*a^11*n^2+23112937/150958080*a^11*n-\ 66985679/127733760*a^12*n^2-367127357/1660538880*a^12*n-112096561/50319360*n^2* a^10-1037207609/103783680*n^3*a^10-105/128*n^11-5/64*n^12-463556473/7188480*n^4 *a^8-11994083/539136*n^4*a^9-753811993/37739520*n^3*a^9-1429555837/12579840*n^5 *a^7+7943164001/207567360*n^3*a^8-18323/1660538880*a^16*n-18323/1660538880*a^15 *n^2-82949/166053888*a^15*n-506426379131/1660538880*n^6*a+975591389/18450432*n^ 7*a+102632415811/1660538880*n^6*a^2-4835193533/5930496*n^5*a^2+9891176045/ 15814656*n^6*a^3-189808205033/1660538880*n^5*a^3+244337581/83026944*n*a^10+ 760244515/13837824*n^2*a^8-999785161/103783680*n*a^8+6215424169/332107776*n^2*a ^9+91397377/18450432*n*a^9-63421301017/138378240*n^3*a^5+29299394119/63866880*n ^4*a^5+125169465/1025024*n^3*a^6+8957262229/23721984*n^4*a^6+60327947/6150144*n ^9-21145/28672*n^10+6362511781/30750720*n^3*a^7-606079027/46126080*n^2*a^7+ 6678891823/207567360*n^5-6676860247/207567360*n^6+210564222299/553512960*n^5*a^ 5-65179143199/830269440*n^5*a^6-1261639391/10063872*n^6*a^6+10104223247/ 830269440*n^4*a^7+10962997099/50319360*n^6*a^4-1113683/6864*n^7*a^4-89672259161 /553512960*n^6*a^5+1611453643/18450432*n^8*a-11625/3584*n^9*a+364883215/1153152 *n^7*a^2+34052075/9225216*n^8*a^2+7107800201/110702592*n^7*a^3-12721165/129024* n^8*a^3-70073/36900864*a^13*n^3-6290323/127733760*a^13*n^2+524453921/1441440*n^ 2*a^3-51177667/960960*n*a^3+14618537/360360*n*a^4+809534527/5322240*n^3*a-\ 1263751/11648*n^2*a-137/3*n*a+776983/174720*n^2*a^2-18502223/288288*n*a^2+ 804865/768768*n*a^6-2965016987/13837824*n^2*a^6-561198893/17297280*n*a^7+ 24377923049/138378240*n^3*a^3-215204446537/207567360*n^4*a^3+857863687/3459456* n^2*a^4-2680206731/4193280*n^3*a^4+2536567109/34594560*n*a^5-15126824513/ 103783680*n^2*a^5+140172576161/415134720*n^4*a-908847761/7907328*n^5*a+ 63379104641/103783680*n^3*a^2-22256344267/415134720*n^4*a^2-445/774144*a^12*n^4 -30313853/276756480*a^12*n^3-3597/32*n^2+529203/8008*n^3+2241295039/34594560*n^ 4-30301/266112*a^11*n^4-211788041/127733760*a^11*n^3-32725526759/830269440*n^7+ 170554903/15814656*n^8-1007/18432*a^10*n^5-12053785/4257792*a^10*n^4-79/8064*a^ 9*n^6-1092667/387072*a^9*n^5-421087/258048*n^6*a^8-14651813/479232*n^5*a^8-\ 1021799/36864*n^6*a^7-85/168*n^7*a^7-25/384*n^8*a^6-1084037/64512*n^7*a^6-845/ 128*n^8*a^5-3437389/36864*n^7*a^5-195/128*n^9*a^4-75/4*n^9*a^3-5/32*n^10*a^3-\ 602141/12288*n^8*a^4-538585/14336*n^9*a^2-665/128*n^10*a^2-11153/504*n+ 1197321563507/1660538880*n^5*a^4-9417813589/23063040*n^4*a^4-1003309/553512960* a^14*n^2-30523/3594240*a^14*n-15/16*n^11*a-535/64*n^10*a and in latex \left( -120+{\frac {18323}{830269440}}\,{a}^{15}+{\frac {82949}{ 83026944}}\,{a}^{14}-{\frac {596971}{1451520}}\,{a}^{10}+{\frac {309959 }{725760}}\,{a}^{11}+{\frac {8527271}{63866880}}\,{a}^{12}+{\frac { 18323}{6918912}}\,{a}^{13}n+{\frac {592717}{4257792}}\,{a}^{11}{n}^{2}+ {\frac {7406857}{7983360}}\,{a}^{11}n+{\frac {445}{387072}}\,{a}^{12}{n }^{2}+{\frac {263399}{3193344}}\,{a}^{12}n+{\frac {264877}{107520}}\,{n }^{2}{a}^{10}+{\frac {5785}{64512}}\,{n}^{3}{a}^{10}-{\frac {368521}{ 40320}}\,{a}^{8}-{\frac {1482059}{241920}}\,{a}^{9}+{\frac {29969}{1260 }}\,{a}^{7}+{\frac {1083331}{63866880}}\,{a}^{13}+{\frac {14755}{6144}} \,{n}^{4}{a}^{8}+{\frac {79}{4032}}\,{n}^{4}{a}^{9}+{\frac {645017}{ 193536}}\,{n}^{3}{a}^{9}+{\frac {395}{448}}\,{n}^{5}{a}^{7}+{\frac { 1955791}{64512}}\,{n}^{3}{a}^{8}-{\frac {4059055}{18432}}\,{n}^{6}a+{ \frac {52775}{3072}}\,{n}^{7}a+{\frac {22325}{3072}}\,{n}^{6}{a}^{2}-{ \frac {123102107}{193536}}\,{n}^{5}{a}^{2}+{\frac {1024415}{6144}}\,{n} ^{6}{a}^{3}-{\frac {848665}{6912}}\,{n}^{5}{a}^{3}+{\frac {12776047}{ 2903040}}\,n{a}^{10}+{\frac {2013193}{55296}}\,{n}^{2}{a}^{8}-{\frac { 4838689}{138240}}\,n{a}^{8}+{\frac {13454419}{829440}}\,{n}^{2}{a}^{9}+ {\frac {1799351}{414720}}\,n{a}^{9}-{\frac {760032671}{1451520}}\,{n}^{ 3}{a}^{5}+{\frac {3013205}{18432}}\,{n}^{4}{a}^{5}+{\frac {892613}{ 41472}}\,{n}^{3}{a}^{6}+{\frac {1351567}{9216}}\,{n}^{4}{a}^{6}+{\frac {155}{64}}\,{n}^{9}+3/16\,{n}^{10}+{\frac {204077}{2016}}\,{n}^{3}{a}^{ 7}-{\frac {5203039}{96768}}\,{n}^{2}{a}^{7}-{\frac {4139}{24}}\,{a}^{4} +{\frac {134143}{2016}}\,{a}^{6}-{\frac {2867}{112}}\,{a}^{5}+{\frac { 110328067}{580608}}\,{n}^{5}-{\frac {21739835}{387072}}\,{n}^{6}+{ \frac {199841}{1536}}\,{n}^{5}{a}^{5}+{\frac {36665}{1536}}\,{n}^{5}{a} ^{6}+{\frac {25}{192}}\,{n}^{6}{a}^{6}+{\frac {1455469}{43008}}\,{n}^{4 }{a}^{7}+{\frac {470365}{6144}}\,{n}^{6}{a}^{4}+{\frac {175}{64}}\,{n}^ {7}{a}^{4}+{\frac {2035}{192}}\,{n}^{6}{a}^{5}+{\frac {315}{16}}\,{n}^{ 8}a+{\frac {15}{8}}\,{n}^{9}a+{\frac {228445}{3072}}\,{n}^{7}{a}^{2}+{ \frac {595}{64}}\,{n}^{8}{a}^{2}+{\frac {6035}{192}}\,{n}^{7}{a}^{3}+{ \frac {5}{16}}\,{n}^{8}{a}^{3}-{\frac {161}{6}}\,{a}^{3}+{\frac {548}{3 }}\,{a}^{2}+{\frac {24743447}{20160}}\,{n}^{2}{a}^{3}-{\frac {11444669} {18480}}\,n{a}^{3}+{\frac {111337351}{443520}}\,n{a}^{4}+{\frac { 16328653}{103680}}\,{n}^{3}a-{\frac {35256667}{27720}}\,{n}^{2}a+{ \frac {59493173}{144144}}\,na-{\frac {27187643}{63360}}\,{n}^{2}{a}^{2} -{\frac {1046262989}{1441440}}\,n{a}^{2}+{\frac {426487}{13440}}\,n{a}^ {6}-{\frac {85871789}{241920}}\,{n}^{2}{a}^{6}-{\frac {12666721}{120960 }}\,n{a}^{7}+{\frac {16413223}{36288}}\,{n}^{3}{a}^{3}-{\frac { 288303509}{290304}}\,{n}^{4}{a}^{3}+{\frac {377926007}{483840}}\,{n}^{2 }{a}^{4}-{\frac {1178697433}{1451520}}\,{n}^{3}{a}^{4}+{\frac {17483839 }{40320}}\,n{a}^{5}-{\frac {123525077}{483840}}\,{n}^{2}{a}^{5}+{\frac {265617553}{290304}}\,{n}^{4}a-{\frac {1657177}{9216}}\,{n}^{5}a+{ \frac {33600853}{20736}}\,{n}^{3}{a}^{2}-{\frac {21688951}{290304}}\,{n }^{4}{a}^{2}-{\frac {40215727}{576576}}\,{n}^{2}-{\frac {1329871843}{ 2661120}}\,{n}^{3}+{\frac {253709107}{1451520}}\,{n}^{4}-{\frac {302195 }{9216}}\,{n}^{7}+{\frac {24605}{6144}}\,{n}^{8}+{\frac {274}{3}}\,a+{ \frac {10103}{24}}\,n+{\frac {28987}{128}}\,{n}^{5}{a}^{4}-{\frac { 21223417}{55296}}\,{n}^{4}{a}^{4} \right) E-{\frac {954251}{14192640}} \,{a}^{13}n-{\frac {189748607}{79073280}}\,{a}^{11}{n}^{2}+{\frac { 23112937}{150958080}}\,{a}^{11}n-{\frac {66985679}{127733760}}\,{a}^{12 }{n}^{2}-{\frac {367127357}{1660538880}}\,{a}^{12}n-{\frac {112096561}{ 50319360}}\,{n}^{2}{a}^{10}-{\frac {1037207609}{103783680}}\,{n}^{3}{a} ^{10}-{\frac {105}{128}}\,{n}^{11}-{\frac {5}{64}}\,{n}^{12}-{\frac { 463556473}{7188480}}\,{n}^{4}{a}^{8}-{\frac {11994083}{539136}}\,{n}^{4 }{a}^{9}-{\frac {753811993}{37739520}}\,{n}^{3}{a}^{9}-{\frac { 1429555837}{12579840}}\,{n}^{5}{a}^{7}+{\frac {7943164001}{207567360}} \,{n}^{3}{a}^{8}-{\frac {18323}{1660538880}}\,{a}^{16}n-{\frac {18323}{ 1660538880}}\,{a}^{15}{n}^{2}-{\frac {82949}{166053888}}\,{a}^{15}n-{ \frac {506426379131}{1660538880}}\,{n}^{6}a+{\frac {975591389}{18450432 }}\,{n}^{7}a+{\frac {102632415811}{1660538880}}\,{n}^{6}{a}^{2}-{\frac {4835193533}{5930496}}\,{n}^{5}{a}^{2}+{\frac {9891176045}{15814656}}\, {n}^{6}{a}^{3}-{\frac {189808205033}{1660538880}}\,{n}^{5}{a}^{3}+{ \frac {244337581}{83026944}}\,n{a}^{10}+{\frac {760244515}{13837824}}\, {n}^{2}{a}^{8}-{\frac {999785161}{103783680}}\,n{a}^{8}+{\frac { 6215424169}{332107776}}\,{n}^{2}{a}^{9}+{\frac {91397377}{18450432}}\,n {a}^{9}-{\frac {63421301017}{138378240}}\,{n}^{3}{a}^{5}+{\frac { 29299394119}{63866880}}\,{n}^{4}{a}^{5}+{\frac {125169465}{1025024}}\,{ n}^{3}{a}^{6}+{\frac {8957262229}{23721984}}\,{n}^{4}{a}^{6}+{\frac { 60327947}{6150144}}\,{n}^{9}-{\frac {21145}{28672}}\,{n}^{10}+{\frac { 6362511781}{30750720}}\,{n}^{3}{a}^{7}-{\frac {606079027}{46126080}}\,{ n}^{2}{a}^{7}+{\frac {6678891823}{207567360}}\,{n}^{5}-{\frac { 6676860247}{207567360}}\,{n}^{6}+{\frac {210564222299}{553512960}}\,{n} ^{5}{a}^{5}-{\frac {65179143199}{830269440}}\,{n}^{5}{a}^{6}-{\frac { 1261639391}{10063872}}\,{n}^{6}{a}^{6}+{\frac {10104223247}{830269440}} \,{n}^{4}{a}^{7}+{\frac {10962997099}{50319360}}\,{n}^{6}{a}^{4}-{ \frac {1113683}{6864}}\,{n}^{7}{a}^{4}-{\frac {89672259161}{553512960}} \,{n}^{6}{a}^{5}+{\frac {1611453643}{18450432}}\,{n}^{8}a-{\frac {11625 }{3584}}\,{n}^{9}a+{\frac {364883215}{1153152}}\,{n}^{7}{a}^{2}+{\frac {34052075}{9225216}}\,{n}^{8}{a}^{2}+{\frac {7107800201}{110702592}}\,{ n}^{7}{a}^{3}-{\frac {12721165}{129024}}\,{n}^{8}{a}^{3}-{\frac {70073} {36900864}}\,{a}^{13}{n}^{3}-{\frac {6290323}{127733760}}\,{a}^{13}{n}^ {2}+{\frac {524453921}{1441440}}\,{n}^{2}{a}^{3}-{\frac {51177667}{ 960960}}\,n{a}^{3}+{\frac {14618537}{360360}}\,n{a}^{4}+{\frac { 809534527}{5322240}}\,{n}^{3}a-{\frac {1263751}{11648}}\,{n}^{2}a-{ \frac {137}{3}}\,na+{\frac {776983}{174720}}\,{n}^{2}{a}^{2}-{\frac { 18502223}{288288}}\,n{a}^{2}+{\frac {804865}{768768}}\,n{a}^{6}-{\frac {2965016987}{13837824}}\,{n}^{2}{a}^{6}-{\frac {561198893}{17297280}}\, n{a}^{7}+{\frac {24377923049}{138378240}}\,{n}^{3}{a}^{3}-{\frac { 215204446537}{207567360}}\,{n}^{4}{a}^{3}+{\frac {857863687}{3459456}} \,{n}^{2}{a}^{4}-{\frac {2680206731}{4193280}}\,{n}^{3}{a}^{4}+{\frac { 2536567109}{34594560}}\,n{a}^{5}-{\frac {15126824513}{103783680}}\,{n}^ {2}{a}^{5}+{\frac {140172576161}{415134720}}\,{n}^{4}a-{\frac { 908847761}{7907328}}\,{n}^{5}a+{\frac {63379104641}{103783680}}\,{n}^{3 }{a}^{2}-{\frac {22256344267}{415134720}}\,{n}^{4}{a}^{2}-{\frac {445}{ 774144}}\,{a}^{12}{n}^{4}-{\frac {30313853}{276756480}}\,{a}^{12}{n}^{3 }-{\frac {3597}{32}}\,{n}^{2}+{\frac {529203}{8008}}\,{n}^{3}+{\frac { 2241295039}{34594560}}\,{n}^{4}-{\frac {30301}{266112}}\,{a}^{11}{n}^{4 }-{\frac {211788041}{127733760}}\,{a}^{11}{n}^{3}-{\frac {32725526759}{ 830269440}}\,{n}^{7}+{\frac {170554903}{15814656}}\,{n}^{8}-{\frac { 1007}{18432}}\,{a}^{10}{n}^{5}-{\frac {12053785}{4257792}}\,{a}^{10}{n} ^{4}-{\frac {79}{8064}}\,{a}^{9}{n}^{6}-{\frac {1092667}{387072}}\,{a}^ {9}{n}^{5}-{\frac {421087}{258048}}\,{n}^{6}{a}^{8}-{\frac {14651813}{ 479232}}\,{n}^{5}{a}^{8}-{\frac {1021799}{36864}}\,{n}^{6}{a}^{7}-{ \frac {85}{168}}\,{n}^{7}{a}^{7}-{\frac {25}{384}}\,{n}^{8}{a}^{6}-{ \frac {1084037}{64512}}\,{n}^{7}{a}^{6}-{\frac {845}{128}}\,{n}^{8}{a}^ {5}-{\frac {3437389}{36864}}\,{n}^{7}{a}^{5}-{\frac {195}{128}}\,{n}^{9 }{a}^{4}-{\frac {75}{4}}\,{n}^{9}{a}^{3}-{\frac {5}{32}}\,{n}^{10}{a}^{ 3}-{\frac {602141}{12288}}\,{n}^{8}{a}^{4}-{\frac {538585}{14336}}\,{n} ^{9}{a}^{2}-{\frac {665}{128}}\,{n}^{10}{a}^{2}-{\frac {11153}{504}}\,n +{\frac {1197321563507}{1660538880}}\,{n}^{5}{a}^{4}-{\frac {9417813589 }{23063040}}\,{n}^{4}{a}^{4}-{\frac {1003309}{553512960}}\,{a}^{14}{n}^ {2}-{\frac {30523}{3594240}}\,{a}^{14}n-{\frac {15}{16}}\,{n}^{11}a-{ \frac {535}{64}}\,{n}^{10}a Theorem Number, 6, : the , 7, -th factorial moment of X is 19 18 2 18 17 2 124799 a n 124799 a n 1345093 a n 7321597 a n - ------------ - ------------- - ------------- - -------------- 239117598720 239117598720 39852933120 59779399680 17 16 3 16 2 15 4 29296769 a n 1132423 a n 2746673 a n 18323 a n - -------------- - -------------- - -------------- - ------------ 34159656960 8856207360 543449088 474439680 15 3 14 4 14 3 13 5 1355828863 a n 8704967 a n 2114527711 a n 356803 a n - ----------------- - -------------- - ----------------- - ------------- 119558799360 738017280 7472424960 63258624 13 4 12 6 12 5 11 6 38657209 a n 445 a n 159026327 a n 1419041 a n - --------------- - ---------- - ---------------- - -------------- 77837760 442368 316293120 4866048 11 5 10 7 10 6 9 8 104491183 a n 19883 a n 163334191 a n 79 a n - ---------------- - ------------ - ---------------- - -------- 9953280 221184 16220160 6912 9 7 8 8 8 7 9 7 699371 a n 1109789 a n 1021459273 a n 585 n a - ------------ - ------------- - ---------------- - --------- 110592 442368 11501568 1024 8 7 10 6 9 6 10 4 1521761 n a 175 n a 2668399 n a 9713249 n a - ------------- - ---------- - ------------- - -------------- 27648 3072 110592 147456 10 5 9 5 11 4 12 13 3675 n a 73185301 n a 1337 n a 2107 n a 21 n a - ----------- - -------------- - ----------- - ---------- - -------- 512 442368 1024 256 32 12 2 11 2 12 3 11 3 4501 n a 3391201 n a 7 n a 20209 n a / - ----------- - -------------- - -------- - ------------ + |720 1024 73728 64 1024 \ 124799 18 1345093 17 8539691 16 2466049 15 + ------------ a + ----------- a + ---------- a + --------- a 119558799360 19926466560 4981616640 113218560 1395513661 14 4390031 10 5493383 11 54578041 12 + ---------- a - ------- a - ------- a - -------- a 9963233280 622080 622080 41057280 42546301 13 86822767 11 2 69014399 11 134247913 12 2 + -------- a n + -------- a n - -------- a n + --------- a n 32348160 3421440 31933440 27371520 26714155 12 328931387 2 10 111132169 3 10 15290971 8 + -------- a n + --------- n a + --------- n a + -------- a 5474304 9953280 1658880 120960 6203111 9 168787 7 973 11 12 118659473 13 + ------- a - ------ a + --- n + 7/64 n + --------- a 103680 1080 512 383201280 20566283 4 8 350559443 4 9 623298079 3 9 1002951 5 7 + -------- n a + --------- n a + --------- n a + ------- n a 55296 3317760 3732480 2048 568350107 3 8 2371181 16 18323 15 2 - --------- n a + ----------- a n + --------- a n 2488320 13284311040 237219840 21070927 15 708099401 6 4785707 7 992534105 6 2 + ---------- a n + --------- n a - ------- n a - --------- n a 2490808320 331776 6912 663552 6675331813 5 2 258494903 6 3 208269013 5 3 857373263 10 + ---------- n a - --------- n a - --------- n a - --------- n a 995328 82944 248832 11612160 4730350279 2 8 756039269 8 1490034337 2 9 8253459817 9 - ---------- n a + --------- n a - ---------- n a - ---------- n a 5806080 2903040 6967296 52254720 6055955719 3 5 21434868377 4 5 955676573 3 6 + ---------- n a - ----------- n a - --------- n a 1244160 4976640 497664 23652929137 4 6 29336885 9 141547 10 8275305217 3 7 - ----------- n a - -------- n + ------ n - ---------- n a 9953280 884736 24576 4354560 19701499 2 7 4 21055 6 1031 5 38882132903 5 - -------- n a + 3710/3 a - ----- a + ---- a - ----------- n 362880 36 12 27371520 5798476513 6 1375782421 5 5 192763235 5 6 3470033 6 6 + ---------- n - ---------- n a + --------- n a + ------- n a 7464960 829440 331776 8192 2152903 4 7 1906205 6 4 42902519 7 4 3542609 6 5 + ------- n a - ------- n a + -------- n a + ------- n a 20736 3456 73728 4608 4952985 8 125195 9 139589123 7 2 32137955 8 2 - ------- n a + ------ n a - --------- n a + -------- n a 16384 3072 110592 294912 2101375 7 3 10554845 8 3 18323 13 3 168097 13 2 3 + ------- n a + -------- n a + ------- a n + ------ a n + 273 a 55296 36864 1976832 399360 2 112698296413 2 3 7909644863 3 96930887 4 - 1176 a - ------------ n a + ---------- n a - -------- n a 10378368 1853280 37440 315775706291 3 53189969533 2 68455916 6698868403 2 2 - ------------ n a + ----------- n a - -------- n a + ---------- n a 124540416 5765760 27027 3144960 819101029 2 3152212829 6 1881170585 2 6 611151931 7 + --------- n a + ---------- n a + ---------- n a + --------- n a 154440 15966720 497664 498960 3614987579 3 3 6967238491 4 3 46790540033 2 4 - ---------- n a + ---------- n a - ----------- n a 2052864 622080 7983360 212499305 3 4 17755629371 5 17788159129 2 5 5640462893 4 + --------- n a - ----------- n a + ----------- n a - ---------- n a 21384 4942080 4561920 760320 7286002103 5 311953312355 3 2 1781692741 4 2 445 12 4 + ---------- n a - ------------ n a + ---------- n a + ------ a n 2488320 22643712 570240 221184 4280389 12 3 4163646947 2 360462421427 3 3375299406581 4 + ------- a n + ---------- n + ------------ n - ------------- n 7299072 5189184 103783680 1868106240 521063 11 4 89276893 11 3 46783981 7 13352107 8 + ------- a n + -------- a n + -------- n - -------- n 1216512 9123840 165888 110592 5785 10 5 26019947 10 4 79 9 6 243053 9 5 + ----- a n + -------- a n + ---- a n + ------ a n 36864 2211840 3456 27648 295775 6 8 16069603 5 8 2827859 6 7 395 7 7 175 8 6 + ------ n a + -------- n a + ------- n a + --- n a + ---- n a 73728 147456 36864 384 1536 685405 7 6 9275 8 5 1170085 7 5 1225 9 4 1645 9 3 + ------ n a + ---- n a + ------- n a + ---- n a + ---- n a 18432 768 4608 512 48 10 3 892185 8 4 1151675 9 2 4109 10 2 + 7/32 n a + ------ n a + ------- n a + ---- n a - 588 a 8192 12288 512 198053 1317084791 5 4 2851496813 4 4 9563717 14 2 - ------ n - ---------- n a + ---------- n a + --------- a n 72 276480 1244160 664215552 13 152442463 14 21 11 4921 10 \ 36585597239 a n + --------- a n + -- n a + ---- n a| E + ----------------- 996323328 16 256 / 59779399680 11 2 11 12 2 292356674297 a n 1960218599 a n 2403056629 a n + ------------------- + ---------------- + ----------------- 7472424960 459841536 1868106240 12 2 10 3 10 10840314601 a n 412793506061 n a 606315158887 n a + ----------------- + ------------------- + ------------------- 2490808320 4981616640 4598415360 11 12 4 8 2913898667 n 203905 n 36508296176483 n a + -------------- - ---------- + -------------------- 253034496 147456 29889699840 4 9 3 9 5 7 11863294234037 n a 6813489478459 n a 224803825853017 n a + -------------------- + ------------------- + --------------------- 59779399680 14944849920 119558799360 3 8 16 15 2 15 6745276039 n a 40812011 a n 235555669 a n 2822620699 a n - ---------------- - -------------- - ---------------- - ---------------- 207567360 3736212480 2717245440 39852933120 6 7 6 2 18055546891 n a 140699459743 n a 6745144321 n a + ---------------- - ----------------- - ---------------- 8339760 158146560 3732480 5 2 6 3 5 3 23251377756949 n a 13850310540263 n a 1965266232469 n a + -------------------- - -------------------- - ------------------- 3736212480 2134978560 2134978560 10 2 8 8 402250392853 n a 193699961953 n a 422094143 n a - ------------------ - ------------------ + -------------- 14944849920 311351040 7983360 2 9 9 3 5 97121574103 n a 3571057153 n a 15950121211873 n a - ----------------- - --------------- + -------------------- 747242496 55351296 4981616640 4 5 3 6 4 6 46106433343 n a 3334313576617 n a 8635167394213 n a - ----------------- - ------------------- - ------------------- 8087040 1660538880 2490808320 9 10 3 7 17829746663 n 1753587559 n 21255711461651 n a - -------------- + -------------- - -------------------- 218972160 63258624 9963233280 2 7 5 6 64800685393 n a 27138954539 n 2157380392109 n - ----------------- - -------------- + ---------------- 415134720 373621248 7472424960 5 5 5 6 6 6 23488290682423 n a 14311166026543 n a 1884202589537 n a - -------------------- + -------------------- + ------------------- 8539914240 5434490880 1067489280 4 7 6 4 7 4 248155883993 n a 134264548967 n a 813834106849 n a + ------------------ - ------------------ + ------------------ 266872320 213497856 258785280 6 5 8 9 31458034126969 n a 2267297667941 n a 1985695475 n a + -------------------- - ------------------ + --------------- 8539914240 2846638080 9371648 7 2 8 2 7 3 2397582669647 n a 876174730489 n a 11317468888729 n a - ------------------- + ------------------ + -------------------- 776355840 1423319040 17079828480 8 3 13 3 13 2 329302390361 n a 182269335461 a n 4458005273 a n + ------------------ - ------------------- - ----------------- 189775872 59779399680 1707982848 2 3 3 4 3 537871504063 n a 334087211 n a 8403220517 n a 54257456299 n a - ------------------ + -------------- - --------------- - ---------------- 207567360 1081080 25945920 41513472 2 2 2 2 6557332255 n a 67578551 n a 177404455 n a + --------------- + 294 n a - -------------- + -------------- 10378368 269568 432432 6 2 6 7 11023187633 n a 261544428773 n a 48436695059 n a + ---------------- + ------------------ + ---------------- 207567360 155675520 177914880 3 3 4 3 2 4 1100030677819 n a 401498808649 n a 14706478697 n a - ------------------- + ------------------ - ----------------- 2490808320 47900160 9729720 3 4 5 2 5 114989479895 n a 12873053251 n a 4044953479 n a + ------------------ - ---------------- + ---------------- 20756736 25945920 2779920 4 5 3 2 51847038617 n a 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15966720}}\,n{a}^{6}+{\frac {1881170585}{497664}}\,{n}^{2}{a}^{6}+{ \frac {611151931}{498960}}\,n{a}^{7}-{\frac {3614987579}{2052864}}\,{n} ^{3}{a}^{3}+{\frac {6967238491}{622080}}\,{n}^{4}{a}^{3}-{\frac { 46790540033}{7983360}}\,{n}^{2}{a}^{4}+{\frac {212499305}{21384}}\,{n}^ {3}{a}^{4}-{\frac {17755629371}{4942080}}\,n{a}^{5}+{\frac {17788159129 }{4561920}}\,{n}^{2}{a}^{5}-{\frac {5640462893}{760320}}\,{n}^{4}a+{ \frac {7286002103}{2488320}}\,{n}^{5}a-{\frac {311953312355}{22643712}} \,{n}^{3}{a}^{2}+{\frac {1781692741}{570240}}\,{n}^{4}{a}^{2}+{\frac { 445}{221184}}\,{a}^{12}{n}^{4}+{\frac {4280389}{7299072}}\,{a}^{12}{n}^ {3}+{\frac {4163646947}{5189184}}\,{n}^{2}+{\frac {360462421427}{ 103783680}}\,{n}^{3}-{\frac {3375299406581}{1868106240}}\,{n}^{4}+{ \frac {521063}{1216512}}\,{a}^{11}{n}^{4}+{\frac {89276893}{9123840}}\, {a}^{11}{n}^{3}+{\frac {46783981}{165888}}\,{n}^{7}-{\frac {13352107}{ 110592}}\,{n}^{8}+{\frac {5785}{36864}}\,{a}^{10}{n}^{5}+{\frac { 26019947}{2211840}}\,{a}^{10}{n}^{4}+{\frac {79}{3456}}\,{a}^{9}{n}^{6} +{\frac {243053}{27648}}\,{a}^{9}{n}^{5}+{\frac {295775}{73728}}\,{n}^{ 6}{a}^{8}+{\frac {16069603}{147456}}\,{n}^{5}{a}^{8}+{\frac {2827859}{ 36864}}\,{n}^{6}{a}^{7}+{\frac {395}{384}}\,{n}^{7}{a}^{7}+{\frac {175} {1536}}\,{n}^{8}{a}^{6}+{\frac {685405}{18432}}\,{n}^{7}{a}^{6}+{\frac {9275}{768}}\,{n}^{8}{a}^{5}+{\frac {1170085}{4608}}\,{n}^{7}{a}^{5}+{ \frac {1225}{512}}\,{n}^{9}{a}^{4}+{\frac {1645}{48}}\,{n}^{9}{a}^{3}+{ \frac {7}{32}}\,{n}^{10}{a}^{3}+{\frac {892185}{8192}}\,{n}^{8}{a}^{4}+ {\frac {1151675}{12288}}\,{n}^{9}{a}^{2}+{\frac {4109}{512}}\,{n}^{10}{ a}^{2}-588\,a-{\frac {198053}{72}}\,n-{\frac {1317084791}{276480}}\,{n} ^{5}{a}^{4}+{\frac {2851496813}{1244160}}\,{n}^{4}{a}^{4}+{\frac { 9563717}{664215552}}\,{a}^{14}{n}^{2}+{\frac {152442463}{996323328}}\,{ a}^{14}n+{\frac {21}{16}}\,{n}^{11}a+{\frac {4921}{256}}\,{n}^{10}a \right) E+{\frac {36585597239}{59779399680}}\,{a}^{13}n+{\frac { 292356674297}{7472424960}}\,{a}^{11}{n}^{2}+{\frac {1960218599}{ 459841536}}\,{a}^{11}n+{\frac {2403056629}{1868106240}}\,{a}^{12}{n}^{2 }+{\frac {10840314601}{2490808320}}\,{a}^{12}n+{\frac {412793506061}{ 4981616640}}\,{n}^{2}{a}^{10}+{\frac {606315158887}{4598415360}}\,{n}^{ 3}{a}^{10}+{\frac {2913898667}{253034496}}\,{n}^{11}-{\frac {203905}{ 147456}}\,{n}^{12}+{\frac {36508296176483}{29889699840}}\,{n}^{4}{a}^{8 }+{\frac {11863294234037}{59779399680}}\,{n}^{4}{a}^{9}+{\frac { 6813489478459}{14944849920}}\,{n}^{3}{a}^{9}+{\frac {224803825853017}{ 119558799360}}\,{n}^{5}{a}^{7}-{\frac {6745276039}{207567360}}\,{n}^{3} {a}^{8}-{\frac {40812011}{3736212480}}\,{a}^{16}n-{\frac {235555669}{ 2717245440}}\,{a}^{15}{n}^{2}-{\frac {2822620699}{39852933120}}\,{a}^{ 15}n+{\frac {18055546891}{8339760}}\,{n}^{6}a-{\frac {140699459743}{ 158146560}}\,{n}^{7}a-{\frac {6745144321}{3732480}}\,{n}^{6}{a}^{2}+{ \frac {23251377756949}{3736212480}}\,{n}^{5}{a}^{2}-{\frac { 13850310540263}{2134978560}}\,{n}^{6}{a}^{3}-{\frac {1965266232469}{ 2134978560}}\,{n}^{5}{a}^{3}-{\frac {402250392853}{14944849920}}\,n{a}^ {10}-{\frac {193699961953}{311351040}}\,{n}^{2}{a}^{8}+{\frac { 422094143}{7983360}}\,n{a}^{8}-{\frac {97121574103}{747242496}}\,{n}^{2 }{a}^{9}-{\frac {3571057153}{55351296}}\,n{a}^{9}+{\frac { 15950121211873}{4981616640}}\,{n}^{3}{a}^{5}-{\frac {46106433343}{ 8087040}}\,{n}^{4}{a}^{5}-{\frac {3334313576617}{1660538880}}\,{n}^{3}{ a}^{6}-{\frac {8635167394213}{2490808320}}\,{n}^{4}{a}^{6}-{\frac { 17829746663}{218972160}}\,{n}^{9}+{\frac {1753587559}{63258624}}\,{n}^{ 10}-{\frac {21255711461651}{9963233280}}\,{n}^{3}{a}^{7}-{\frac { 64800685393}{415134720}}\,{n}^{2}{a}^{7}-{\frac {27138954539}{373621248 }}\,{n}^{5}+{\frac {2157380392109}{7472424960}}\,{n}^{6}-{\frac { 23488290682423}{8539914240}}\,{n}^{5}{a}^{5}+{\frac {14311166026543}{ 5434490880}}\,{n}^{5}{a}^{6}+{\frac {1884202589537}{1067489280}}\,{n}^{ 6}{a}^{6}+{\frac {248155883993}{266872320}}\,{n}^{4}{a}^{7}-{\frac { 134264548967}{213497856}}\,{n}^{6}{a}^{4}+{\frac {813834106849}{ 258785280}}\,{n}^{7}{a}^{4}+{\frac {31458034126969}{8539914240}}\,{n}^{ 6}{a}^{5}-{\frac {2267297667941}{2846638080}}\,{n}^{8}a+{\frac { 1985695475}{9371648}}\,{n}^{9}a-{\frac {2397582669647}{776355840}}\,{n} ^{7}{a}^{2}+{\frac {876174730489}{1423319040}}\,{n}^{8}{a}^{2}+{\frac { 11317468888729}{17079828480}}\,{n}^{7}{a}^{3}+{\frac {329302390361}{ 189775872}}\,{n}^{8}{a}^{3}-{\frac {182269335461}{59779399680}}\,{a}^{ 13}{n}^{3}-{\frac {4458005273}{1707982848}}\,{a}^{13}{n}^{2}-{\frac { 537871504063}{207567360}}\,{n}^{2}{a}^{3}+{\frac {334087211}{1081080}} \,n{a}^{3}-{\frac {8403220517}{25945920}}\,n{a}^{4}-{\frac {54257456299 }{41513472}}\,{n}^{3}a+{\frac {6557332255}{10378368}}\,{n}^{2}a+294\,na -{\frac {67578551}{269568}}\,{n}^{2}{a}^{2}+{\frac {177404455}{432432}} \,n{a}^{2}+{\frac {11023187633}{207567360}}\,n{a}^{6}+{\frac { 261544428773}{155675520}}\,{n}^{2}{a}^{6}+{\frac {48436695059}{ 177914880}}\,n{a}^{7}-{\frac {1100030677819}{2490808320}}\,{n}^{3}{a}^{ 3}+{\frac {401498808649}{47900160}}\,{n}^{4}{a}^{3}-{\frac {14706478697 }{9729720}}\,{n}^{2}{a}^{4}+{\frac {114989479895}{20756736}}\,{n}^{3}{a }^{4}-{\frac {12873053251}{25945920}}\,n{a}^{5}+{\frac {4044953479}{ 2779920}}\,{n}^{2}{a}^{5}-{\frac {51847038617}{26127360}}\,{n}^{4}a+{ \frac {1327863762913}{996323328}}\,{n}^{5}a-{\frac {5088408111263}{ 1245404160}}\,{n}^{3}{a}^{2}+{\frac {5285398937153}{3736212480}}\,{n}^{ 4}{a}^{2}-{\frac {12223316141}{1707982848}}\,{a}^{12}{n}^{4}-{\frac { 126380068037}{8539914240}}\,{a}^{12}{n}^{3}+{\frac {23037}{32}}\,{n}^{2 }-{\frac {1438864657}{2594592}}\,{n}^{3}-{\frac {1203989153}{2661120}} \,{n}^{4}-{\frac {34496942213}{776355840}}\,{a}^{11}{n}^{4}-{\frac { 1912569086983}{119558799360}}\,{a}^{11}{n}^{3}+{\frac {13095880043}{ 53374464}}\,{n}^{7}-{\frac {4366478747711}{29889699840}}\,{n}^{8}-{ \frac {4262079635}{51757056}}\,{a}^{10}{n}^{5}-{\frac {408092449129}{ 4269957120}}\,{a}^{10}{n}^{4}-{\frac {101698397}{995328}}\,{a}^{9}{n}^{ 6}-{\frac {129610816951}{517570560}}\,{a}^{9}{n}^{5}-{\frac {3786783653 }{9584640}}\,{n}^{6}{a}^{8}+{\frac {510831972557}{8539914240}}\,{n}^{5} {a}^{8}-{\frac {22342725547}{79073280}}\,{n}^{6}{a}^{7}-{\frac { 167481957857}{402554880}}\,{n}^{7}{a}^{7}-{\frac {590477935}{1916928}} \,{n}^{8}{a}^{6}-{\frac {3693300131}{6589440}}\,{n}^{7}{a}^{6}-{\frac { 14800436633}{26357760}}\,{n}^{8}{a}^{5}+{\frac {948262630309}{948879360 }}\,{n}^{7}{a}^{5}-{\frac {22846126079}{63258624}}\,{n}^{9}{a}^{4}-{ \frac {444346099}{84344832}}\,{n}^{9}{a}^{3}-{\frac {69651035}{442368}} \,{n}^{10}{a}^{3}+{\frac {91660108979}{316293120}}\,{n}^{8}{a}^{4}+{ \frac {77713369585}{126517248}}\,{n}^{9}{a}^{2}-{\frac {8935648255}{ 253034496}}\,{n}^{10}{a}^{2}+{\frac {3383}{24}}\,n-{\frac { 17134928390101}{2134978560}}\,{n}^{5}{a}^{4}+{\frac {446921945837}{ 233513280}}\,{n}^{4}{a}^{4}-{\frac {693}{1024}}\,{n}^{13}-{\frac {3}{64 }}\,{n}^{14}-{\frac {830381119}{1149603840}}\,{a}^{14}{n}^{2}-{\frac { 3271785527}{19926466560}}\,{a}^{14}n-{\frac {858749}{73728}}\,{n}^{11}a +{\frac {895744445}{7028736}}\,{n}^{10}a Theorem Number, 7, : the , 8, -th factorial moment of X is 21 2 21 22 13 2 770951 a n 26487497 a n 770951 a n 905755 n a - -------------- - -------------- - -------------- - ------------- 36862151639040 14481559572480 36862151639040 18432 13 3 14 3 11 8 11 7 12 6 7091 n a 7 n a 934225 a n 3799759307 a n 35 n a - ----------- - -------- - ------------- - ----------------- - --------- 384 96 1824768 109486080 768 11 6 10 9 10 8 19 3 793777 n a 713 a n 830076271 a n 11132929 a n - ------------- - ---------- - ---------------- - --------------- 27648 6144 36495360 1609062174720 19 2 12 5 11 5 13 4 41296957183 a n 1757 n a 8665937 n a 791 n a - ------------------ - ----------- - -------------- - ---------- 106706228428800 256 36864 768 12 4 8 10 8 9 7 11 2769515 n a 344351 a n 1548034681 a n 215 a n - -------------- - ------------- - ---------------- - ---------- 36864 110592 8626176 384 7 10 3093173 a n / 26487497 20 8480461 19 - -------------- + |-5040 + ------------- a + ------------ a n 36864 \ 7240779786240 877670277120 124799 18 2 70526147 18 46942897 17 2 + ----------- a n + ------------ a n + ----------- a n 29889699840 108883906560 42343741440 8762340439 17 2371181 16 3 40270921 16 2 + ------------ a n + ---------- a n + --------- a n 508124897280 3321077760 830269440 18323 15 4 171170533 15 3 8402431 14 4 + --------- a n + ---------- a n + --------- a n 118609920 2490808320 166053888 114834283 14 3 18323 13 5 42022597 13 4 445 12 6 + --------- a n + ------ a n + -------- a n + ------ a n 62270208 988416 18247680 165888 3226793 12 5 1491535 11 6 738571657 11 5 5785 10 7 + ------- a n + ------- a n + --------- a n + ----- a n 1824768 1824768 18247680 27648 50790397 10 6 79 9 8 1299407 9 7 96245 8 8 + -------- a n 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267886544301721711}{2654952588288000}}\,{a}^{14}n-{\frac { 551376631035519565}{54922655563776}}\,{n}^{11}a+{\frac { 515136966381946259}{42248196587520}}\,{n}^{10}a Theorem Number, 9, : the , 10, -th factorial moment of X is 10 13 10 12 19 7 19 6 18619 a n 297277357 a n 160082645 a n 96659305437883 a n - ------------- - ----------------- - ---------------- - --------------------- 147456 5406720 2574499479552 3844585889464320 16 9 16 8 15 5 16 5 22459 a n 360122443315373 a n 91371041 n a 4995 n a - ------------ - ---------------------- - --------------- - ----------- 15138816 485631901827072 294912 1024 15 10 15 9 21 6 18323 a n 2763018103 a n 770951 a n - ------------- - ----------------- - ------------- 126517248 7970586624 3276635701248 21 5 28 26 2 58531146039239 a n 21026753291 a n 2187370618637 a n - --------------------- - --------------------- - --------------------- 169161779136430080 945445183593507717120 189089036718701543424 26 22 5 6 16 6 15 35188900893647 a n 33614866849 a n 25 a n 2098367 a n - --------------------- - ------------------ - --------- - -------------- 181816381460289945600 25374266870464512 1024 73728 14 10 14 9 18 8 3462257 a n 1289814436590187 a n 124799 a n - --------------- - ----------------------- - ------------- 32800768 67845633343488 15941173248 18 7 18 3 17 3 25 37639900460165 a n 15 n a 1635 n a 1822773728190251 a n - --------------------- - --------- - ----------- - ---------------------- 2306751533678592 512 128 278072112821619916800 24 2 24 51158074857743641 a n 213843396476380127 a n - ------------------------ - ------------------------ 945445183593507717120 1575741972655846195200 23 2 23 947905445308202723 a n 120060568764256277 a n - ------------------------- - ------------------------ 675317988281076940800 68510520550254182400 20 6 20 5 19 18 932660317751 a n 15475783530503129 a n 45 n a 2445 n a - ------------------- - ------------------------ - -------- - ---------- 4613503067357184 622822914093219840 256 512 18 2 17 2 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