Explicit (Efficient!) Exact Expressions for the Factorial Moments of the AREA 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 AREA 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, 2, : the , 2, -th factorial moment of X is 2 2 4 2 3 3 3 2 5 n a n a n a 5 n n a (-1/3 a - 2/3 a - 1/3 a - 1 - 7/3 n) E + ------- + ---- - ---- + ---- + ---- 12 6 4 12 6 2 2 n 5 n a n a 4 n - ---- + ------ - --- - ---- 12 6 3 3 and in Maple format (-1/3*a^3-2/3*a^2-1/3*a-1-7/3*n)*E+5/12*n^2*a^2+1/6*n*a^4-1/4*n*a^2+5/12*n^3+1/ 6*n*a^3-1/12*n+5/6*n^2*a-1/3*n*a-4/3*n^2 and in latex \left( -1/3\,{a}^{3}-2/3\,{a}^{2}-1/3\,a-1-7/3\,n \right) E+{\frac {5} {12}}\,{n}^{2}{a}^{2}+1/6\,n{a}^{4}-1/4\,n{a}^{2}+{\frac {5}{12}}\,{n}^ {3}+1/6\,n{a}^{3}-1/12\,n+5/6\,{n}^{2}a-1/3\,na-4/3\,{n}^{2} Theorem Number, 3, : the , 3, -th factorial moment of X is / 6 2 15 3 19 4 3 19 5 193 2 |5/96 a + 2 a + -- n + a + 27/4 n + -- a + 3/2 a + -- a + 2 + --- n \ 32 24 48 48 25 15 2 145 2 15 2 2 43 3 4\ + -- n a + -- n a + --- n a + -- n a + -- n a + 3/32 n a | E 24 16 48 32 24 / 5 6 7 2 2 3 3 11 n a 11 n a 5 n a 5 n a 125 n a 3 n a - ------- - ------- + n/4 - ------ + ------- - -------- - ------ 64 64 192 32 64 32 3 2 2 3 3 2 3 2 4 4 75 n a 137 n a 25 n a 5 n a 2 191 n a 175 n - -------- - -------- - -------- - ------- + 4 n - --------- - ------ 64 96 192 8 192 192 2 5 4 3 2 7 n a 5 n a 57 n 37 n a - ------- - ------ + ----- + ------- + n a 96 24 32 48 and in Maple format (5/96*a^6+2*a^2+15/32*n^3+a+27/4*n+19/24*a^4+3/2*a^3+19/48*a^5+2+193/48*n^2+25/ 24*n*a+15/16*n^2*a+145/48*n*a^2+15/32*n^2*a^2+43/24*n*a^3+3/32*n*a^4)*E-11/64*n *a^5-11/64*n*a^6+1/4*n-5/192*n*a^7+5/32*n^2*a^2-125/64*n^3*a-3/32*n*a^3-75/64*n ^3*a^2-137/96*n^2*a-25/192*n^3*a^3-5/8*n^2*a^3+4*n^2-191/192*n^2*a^4-175/192*n^ 4-7/96*n^2*a^5-5/24*n*a^4+57/32*n^3+37/48*n*a^2+n*a and in latex \left( {\frac {5}{96}}\,{a}^{6}+2\,{a}^{2}+{\frac {15}{32}}\,{n}^{3}+a +{\frac {27}{4}}\,n+{\frac {19}{24}}\,{a}^{4}+3/2\,{a}^{3}+{\frac {19}{ 48}}\,{a}^{5}+2+{\frac {193}{48}}\,{n}^{2}+{\frac {25}{24}}\,na+{\frac {15}{16}}\,{n}^{2}a+{\frac {145}{48}}\,n{a}^{2}+{\frac {15}{32}}\,{n}^{ 2}{a}^{2}+{\frac {43}{24}}\,n{a}^{3}+{\frac {3}{32}}\,n{a}^{4} \right) E-{\frac {11}{64}}\,n{a}^{5}-{\frac {11}{64}}\,n{a}^{6}+1/4\,n-{\frac { 5}{192}}\,n{a}^{7}+{\frac {5}{32}}\,{n}^{2}{a}^{2}-{\frac {125}{64}}\,{ n}^{3}a-{\frac {3}{32}}\,n{a}^{3}-{\frac {75}{64}}\,{n}^{3}{a}^{2}-{ \frac {137}{96}}\,{n}^{2}a-{\frac {25}{192}}\,{n}^{3}{a}^{3}-5/8\,{n}^{ 2}{a}^{3}+4\,{n}^{2}-{\frac {191}{192}}\,{n}^{2}{a}^{4}-{\frac {175}{ 192}}\,{n}^{4}-{\frac {7}{96}}\,{n}^{2}{a}^{5}-{\frac {5}{24}}\,n{a}^{4 }+{\frac {57}{32}}\,{n}^{3}+{\frac {37}{48}}\,n{a}^{2}+na Theorem Number, 4, : the , 4, -th factorial moment of X is / 331 7 11 8 79 9 2353 5 89 6 7 75 3 |-6 - --- a - --- a - ----- a - ---- n a - --- n a - 3/112 n a - -- n a \ 720 126 15120 720 144 16 45 3 2 3 3 1061 2 3 27 2 4 2 5 1717 2 2 - -- n a - 5/16 n a - ---- n a - -- n a - 3/16 n a - ---- n a 16 144 16 144 14767 2 941 3 4133 4 2 3 2 - ----- n a - --- n a - ---- n a - 22/3 a - 33/5 a - 68/9 n a 840 72 720 853 57863 2 3809 3 35 4 371 5 907 6 289 4 - --- n a - ----- n - ---- n - -- n - --- a - --- a - --- a - 11/3 a 140 2520 432 16 120 720 60 5 6 7 3 3 2 \ 1657 n a 1273 n a 781 n a 280151 n a 4241 n a - 145/6 n| E + --------- + --------- + -------- + ----------- + ---------- / 3360 1680 2160 30240 3024 3 3 2 3 2 4 2 5 2 2 12311 n a 487 n a 21485 n a 20351 n a 9029 n a + ----------- + --------- + ----------- + ----------- - ---------- 7560 360 6048 15120 2520 2 3 4 2 4 2419 n a 901 n a 241 n a 2903 n a 11 n a 3233 n a - --------- - -------- - -------- + --------- - ------ + --------- 840 1680 5040 1008 3 1890 5 4 2 5 2 4 3 3 4 221 n a 16925 n a 221 n a 5155 n a 119183 n a + -------- + ----------- + --------- + ---------- + ------------ 252 6048 504 3024 30240 4 4 3 5 2 6 3 6 2 7 8 221 n a 4361 n a 6361 n a 3161 n a 5051 n a 347 n a + --------- + ---------- + ---------- + ---------- + ---------- + -------- 720 4320 4320 30240 15120 1890 2 8 9 10 2 3 4 121 n a 1241 n a 79 n a 703 n 21743 n 2203 n 109 n + --------- + --------- + -------- - ------ - -------- + ------- - ----- 7560 30240 30240 48 1680 10080 120 5 6 23957 n 221 n + -------- + ------ 30240 1008 and in Maple format (-6-331/720*a^7-11/126*a^8-79/15120*a^9-2353/720*n*a^5-89/144*n*a^6-3/112*n*a^7 -75/16*n^3*a-45/16*n^3*a^2-5/16*n^3*a^3-1061/144*n^2*a^3-27/16*n^2*a^4-3/16*n^2 *a^5-1717/144*n^2*a^2-14767/840*n*a^2-941/72*n*a^3-4133/720*n*a^4-22/3*a^2-33/5 *a^3-68/9*n^2*a-853/140*n*a-57863/2520*n^2-3809/432*n^3-35/16*n^4-371/120*a^5-\ 907/720*a^6-289/60*a^4-11/3*a-145/6*n)*E+1657/3360*n*a^5+1273/1680*n*a^6+781/ 2160*n*a^7+280151/30240*n^3*a+4241/3024*n^3*a^2+12311/7560*n^3*a^3+487/360*n^2* a^3+21485/6048*n^2*a^4+20351/15120*n^2*a^5-9029/2520*n^2*a^2-2419/840*n*a^2-901 /1680*n*a^3-241/5040*n*a^4+2903/1008*n^2*a-11/3*n*a+3233/1890*n^4*a+221/252*n^5 *a+16925/6048*n^4*a^2+221/504*n^5*a^2+5155/3024*n^4*a^3+119183/30240*n^3*a^4+ 221/720*n^4*a^4+4361/4320*n^3*a^5+6361/4320*n^2*a^6+3161/30240*n^3*a^6+5051/ 15120*n^2*a^7+347/1890*n*a^8+121/7560*n^2*a^8+1241/30240*n*a^9+79/30240*n*a^10-\ 703/48*n^2-21743/1680*n^3+2203/10080*n^4-109/120*n+23957/30240*n^5+221/1008*n^6 and in latex \left( -6-{\frac {331}{720}}\,{a}^{7}-{\frac {11}{126}}\,{a}^{8}-{ \frac {79}{15120}}\,{a}^{9}-{\frac {2353}{720}}\,n{a}^{5}-{\frac {89}{ 144}}\,n{a}^{6}-{\frac {3}{112}}\,n{a}^{7}-{\frac {75}{16}}\,{n}^{3}a-{ \frac {45}{16}}\,{n}^{3}{a}^{2}-{\frac {5}{16}}\,{n}^{3}{a}^{3}-{\frac {1061}{144}}\,{n}^{2}{a}^{3}-{\frac {27}{16}}\,{n}^{2}{a}^{4}-3/16\,{n} ^{2}{a}^{5}-{\frac {1717}{144}}\,{n}^{2}{a}^{2}-{\frac {14767}{840}}\,n {a}^{2}-{\frac {941}{72}}\,n{a}^{3}-{\frac {4133}{720}}\,n{a}^{4}-{ \frac {22}{3}}\,{a}^{2}-{\frac {33}{5}}\,{a}^{3}-{\frac {68}{9}}\,{n}^{ 2}a-{\frac {853}{140}}\,na-{\frac {57863}{2520}}\,{n}^{2}-{\frac {3809} {432}}\,{n}^{3}-{\frac {35}{16}}\,{n}^{4}-{\frac {371}{120}}\,{a}^{5}-{ \frac {907}{720}}\,{a}^{6}-{\frac {289}{60}}\,{a}^{4}-11/3\,a-{\frac { 145}{6}}\,n \right) E+{\frac {1657}{3360}}\,n{a}^{5}+{\frac {1273}{1680 }}\,n{a}^{6}+{\frac {781}{2160}}\,n{a}^{7}+{\frac {280151}{30240}}\,{n} ^{3}a+{\frac {4241}{3024}}\,{n}^{3}{a}^{2}+{\frac {12311}{7560}}\,{n}^{ 3}{a}^{3}+{\frac {487}{360}}\,{n}^{2}{a}^{3}+{\frac {21485}{6048}}\,{n} ^{2}{a}^{4}+{\frac {20351}{15120}}\,{n}^{2}{a}^{5}-{\frac {9029}{2520}} \,{n}^{2}{a}^{2}-{\frac {2419}{840}}\,n{a}^{2}-{\frac {901}{1680}}\,n{a }^{3}-{\frac {241}{5040}}\,n{a}^{4}+{\frac {2903}{1008}}\,{n}^{2}a-11/3 \,na+{\frac {3233}{1890}}\,{n}^{4}a+{\frac {221}{252}}\,{n}^{5}a+{ \frac {16925}{6048}}\,{n}^{4}{a}^{2}+{\frac {221}{504}}\,{n}^{5}{a}^{2} +{\frac {5155}{3024}}\,{n}^{4}{a}^{3}+{\frac {119183}{30240}}\,{n}^{3}{ a}^{4}+{\frac {221}{720}}\,{n}^{4}{a}^{4}+{\frac {4361}{4320}}\,{n}^{3} {a}^{5}+{\frac {6361}{4320}}\,{n}^{2}{a}^{6}+{\frac {3161}{30240}}\,{n} ^{3}{a}^{6}+{\frac {5051}{15120}}\,{n}^{2}{a}^{7}+{\frac {347}{1890}}\, n{a}^{8}+{\frac {121}{7560}}\,{n}^{2}{a}^{8}+{\frac {1241}{30240}}\,n{a }^{9}+{\frac {79}{30240}}\,n{a}^{10}-{\frac {703}{48}}\,{n}^{2}-{\frac {21743}{1680}}\,{n}^{3}+{\frac {2203}{10080}}\,{n}^{4}-{\frac {109}{120 }}\,n+{\frac {23957}{30240}}\,{n}^{5}+{\frac {221}{1008}}\,{n}^{6} Theorem Number, 5, : the , 5, -th factorial moment of X is 6 6 2 6 3 5 3 5 4 3069505 n a 105845 n a 105845 n a 279181 n a 1983707 n a - ------------ - ------------ - ------------ - ------------ - ------------- 774144 48384 774144 36864 774144 5 5 4 5 4 6 4 7 3 7 17101 n a 38388479 n a 1609 n a 31151 n a 207047 n a - ----------- - -------------- - ---------- - ----------- - ------------ 110592 5806080 1152 387072 71680 3 8 3 9 2 9 2 10 2 11 922351 n a 23227 n a 265493 n a 9481 n a 4819 n a - ------------ - ----------- - ------------ - ----------- - ----------- 2322432 1161216 430080 145152 2322432 11 12 13 601271 n a 12583 n a 445 n a / 167159 10 3257 11 - ------------ - ----------- - --------- + |24 + ------- a + ------ a 11612160 2322432 2322432 \ 1451520 290304 445 12 2417 7 48257 8 140039 9 2242867 5 16739 6 + ------- a + ---- a + ----- a + ------ a + ------- n a + ----- n a 1161216 432 24192 241920 60480 1152 129907 7 504085 3 1019905 3 2 300235 3 3 70587 2 3 + ------ n a + ------ n a + ------- n a + ------ n a + ----- n a 24192 10368 20736 10368 896 520801 2 4 8581 2 5 2513177 2 2 67381 2 57209 3 + ------ n a + ---- n a + ------- n a + ----- n a + ----- n a 13824 432 24192 672 672 6547627 4 2 3 620267 2 58619 3545 4 + ------- n a + 100/3 a + 67/2 a + ------ n a + ----- n a + ---- n a 120960 12096 1680 256 565 5 6075 4 2 565 5 2 1875 4 3 18475 3 4 + --- n a + ---- n a + ---- n a + ---- n a + ----- n a 512 512 1024 512 1536 565 4 4 1805 3 5 19145 2 6 65 3 6 185 2 7 + ---- n a + ---- n a + ----- n a + --- n a + --- n a 2048 768 3456 512 256 245197 8 75 2 8 35983 9 27 10 2604667 2 + ------ n a + ---- n a + ------ n a + ---- n a + ------- n 193536 2048 290304 7168 20160 505357 3 1228465 4 1001 5 16477 6 4 2591 + ------ n + ------- n + ---- a + ----- a + 227/8 a + 50/3 a + ---- n 6912 41472 48 1440 24 5 6 7 6345 5 565 6\ 36763 n a 60749 n a 54949 n a + ---- n + ---- n | E - ---------- - ---------- - ---------- 1024 2048 / 48384 20160 24192 3 3 2 3 3 2 3 2 4 4321325 n a 2703721 n a 5060597 n a 5441 n a 40657 n a - ------------ + ------------- - ------------- + ---------- - ----------- 96768 207360 967680 4032 3840 2 5 2 2 2 3 4 5335139 n a 310411 n a 14873 n a 11661 n a 3047 n a - ------------- + ------------ + ---------- + ---------- + --------- 725760 12096 1120 2240 960 2 4 5 4 2 226349 n a 50 n a 7414199 n a 6936221 n a 16466053 n a - ----------- + ------ - ------------ - ------------ - -------------- 40320 3 580608 1161216 1451520 5 2 4 3 3 4 4 4 11157193 n a 289837 n a 5222461 n a 41554217 n a - -------------- - ------------ - ------------- - -------------- 1161216 20736 193536 2903040 3 5 2 6 3 6 2 7 8 3587317 n a 202389 n a 8275991 n a 33945 n a 144857 n a - ------------- - ------------ - ------------- - ----------- - ----------- 322560 17920 967680 7168 90720 2 8 9 10 2 3 1419683 n a 1278173 n a 263161 n a 1595 n 1056967 n - ------------- - ------------ - ------------ + ------- + ---------- 645120 1935360 1161216 24 13440 4 7 5 6 972641 n 49 n 105845 n 5189 n 1377331 n + --------- + ---- - --------- + ------- - ---------- 53760 12 110592 107520 1161216 and in Maple format -3069505/774144*n^6*a-105845/48384*n^6*a^2-105845/774144*n^6*a^3-279181/36864*n ^5*a^3-1983707/774144*n^5*a^4-17101/110592*n^5*a^5-38388479/5806080*n^4*a^5-\ 1609/1152*n^4*a^6-31151/387072*n^4*a^7-207047/71680*n^3*a^7-922351/2322432*n^3* a^8-23227/1161216*n^3*a^9-265493/430080*n^2*a^9-9481/145152*n^2*a^10-4819/ 2322432*n^2*a^11-601271/11612160*n*a^11-12583/2322432*n*a^12-445/2322432*n*a^13 +(24+167159/1451520*a^10+3257/290304*a^11+445/1161216*a^12+2417/432*a^7+48257/ 24192*a^8+140039/241920*a^9+2242867/60480*n*a^5+16739/1152*n*a^6+129907/24192*n *a^7+504085/10368*n^3*a+1019905/20736*n^3*a^2+300235/10368*n^3*a^3+70587/896*n^ 2*a^3+520801/13824*n^2*a^4+8581/432*n^2*a^5+2513177/24192*n^2*a^2+67381/672*n*a ^2+57209/672*n*a^3+6547627/120960*n*a^4+100/3*a^2+67/2*a^3+620267/12096*n^2*a+ 58619/1680*n*a+3545/256*n^4*a+565/512*n^5*a+6075/512*n^4*a^2+565/1024*n^5*a^2+ 1875/512*n^4*a^3+18475/1536*n^3*a^4+565/2048*n^4*a^4+1805/768*n^3*a^5+19145/ 3456*n^2*a^6+65/512*n^3*a^6+185/256*n^2*a^7+245197/193536*n*a^8+75/2048*n^2*a^8 +35983/290304*n*a^9+27/7168*n*a^10+2604667/20160*n^2+505357/6912*n^3+1228465/ 41472*n^4+1001/48*a^5+16477/1440*a^6+227/8*a^4+50/3*a+2591/24*n+6345/1024*n^5+ 565/2048*n^6)*E-36763/48384*n*a^5-60749/20160*n*a^6-54949/24192*n*a^7-4321325/ 96768*n^3*a+2703721/207360*n^3*a^2-5060597/967680*n^3*a^3+5441/4032*n^2*a^3-\ 40657/3840*n^2*a^4-5335139/725760*n^2*a^5+310411/12096*n^2*a^2+14873/1120*n*a^2 +11661/2240*n*a^3+3047/960*n*a^4-226349/40320*n^2*a+50/3*n*a-7414199/580608*n^4 *a-6936221/1161216*n^5*a-16466053/1451520*n^4*a^2-11157193/1161216*n^5*a^2-\ 289837/20736*n^4*a^3-5222461/193536*n^3*a^4-41554217/2903040*n^4*a^4-3587317/ 322560*n^3*a^5-202389/17920*n^2*a^6-8275991/967680*n^3*a^6-33945/7168*n^2*a^7-\ 144857/90720*n*a^8-1419683/645120*n^2*a^8-1278173/1935360*n*a^9-263161/1161216* n*a^10+1595/24*n^2+1056967/13440*n^3+972641/53760*n^4+49/12*n-105845/110592*n^7 +5189/107520*n^5-1377331/1161216*n^6 and in latex -{\frac {3069505}{774144}}\,{n}^{6}a-{\frac {105845}{48384}}\,{n}^{6}{a }^{2}-{\frac {105845}{774144}}\,{n}^{6}{a}^{3}-{\frac {279181}{36864}} \,{n}^{5}{a}^{3}-{\frac {1983707}{774144}}\,{n}^{5}{a}^{4}-{\frac { 17101}{110592}}\,{n}^{5}{a}^{5}-{\frac {38388479}{5806080}}\,{n}^{4}{a} ^{5}-{\frac {1609}{1152}}\,{n}^{4}{a}^{6}-{\frac {31151}{387072}}\,{n}^ {4}{a}^{7}-{\frac {207047}{71680}}\,{n}^{3}{a}^{7}-{\frac {922351}{ 2322432}}\,{n}^{3}{a}^{8}-{\frac {23227}{1161216}}\,{n}^{3}{a}^{9}-{ \frac {265493}{430080}}\,{n}^{2}{a}^{9}-{\frac {9481}{145152}}\,{n}^{2} {a}^{10}-{\frac {4819}{2322432}}\,{n}^{2}{a}^{11}-{\frac {601271}{ 11612160}}\,n{a}^{11}-{\frac {12583}{2322432}}\,n{a}^{12}-{\frac {445}{ 2322432}}\,n{a}^{13}+ \left( 24+{\frac {167159}{1451520}}\,{a}^{10}+{ \frac {3257}{290304}}\,{a}^{11}+{\frac {445}{1161216}}\,{a}^{12}+{ \frac {2417}{432}}\,{a}^{7}+{\frac {48257}{24192}}\,{a}^{8}+{\frac { 140039}{241920}}\,{a}^{9}+{\frac {2242867}{60480}}\,n{a}^{5}+{\frac { 16739}{1152}}\,n{a}^{6}+{\frac {129907}{24192}}\,n{a}^{7}+{\frac { 504085}{10368}}\,{n}^{3}a+{\frac {1019905}{20736}}\,{n}^{3}{a}^{2}+{ \frac {300235}{10368}}\,{n}^{3}{a}^{3}+{\frac {70587}{896}}\,{n}^{2}{a} ^{3}+{\frac {520801}{13824}}\,{n}^{2}{a}^{4}+{\frac {8581}{432}}\,{n}^{ 2}{a}^{5}+{\frac {2513177}{24192}}\,{n}^{2}{a}^{2}+{\frac {67381}{672}} \,n{a}^{2}+{\frac {57209}{672}}\,n{a}^{3}+{\frac {6547627}{120960}}\,n{ a}^{4}+{\frac {100}{3}}\,{a}^{2}+{\frac {67}{2}}\,{a}^{3}+{\frac { 620267}{12096}}\,{n}^{2}a+{\frac {58619}{1680}}\,na+{\frac {3545}{256}} \,{n}^{4}a+{\frac {565}{512}}\,{n}^{5}a+{\frac {6075}{512}}\,{n}^{4}{a} ^{2}+{\frac {565}{1024}}\,{n}^{5}{a}^{2}+{\frac {1875}{512}}\,{n}^{4}{a }^{3}+{\frac {18475}{1536}}\,{n}^{3}{a}^{4}+{\frac {565}{2048}}\,{n}^{4 }{a}^{4}+{\frac {1805}{768}}\,{n}^{3}{a}^{5}+{\frac {19145}{3456}}\,{n} ^{2}{a}^{6}+{\frac {65}{512}}\,{n}^{3}{a}^{6}+{\frac {185}{256}}\,{n}^{ 2}{a}^{7}+{\frac {245197}{193536}}\,n{a}^{8}+{\frac {75}{2048}}\,{n}^{2 }{a}^{8}+{\frac {35983}{290304}}\,n{a}^{9}+{\frac {27}{7168}}\,n{a}^{10 }+{\frac {2604667}{20160}}\,{n}^{2}+{\frac {505357}{6912}}\,{n}^{3}+{ \frac {1228465}{41472}}\,{n}^{4}+{\frac {1001}{48}}\,{a}^{5}+{\frac { 16477}{1440}}\,{a}^{6}+{\frac {227}{8}}\,{a}^{4}+{\frac {50}{3}}\,a+{ \frac {2591}{24}}\,n+{\frac {6345}{1024}}\,{n}^{5}+{\frac {565}{2048}} \,{n}^{6} \right) E-{\frac {36763}{48384}}\,n{a}^{5}-{\frac {60749}{ 20160}}\,n{a}^{6}-{\frac {54949}{24192}}\,n{a}^{7}-{\frac {4321325}{ 96768}}\,{n}^{3}a+{\frac {2703721}{207360}}\,{n}^{3}{a}^{2}-{\frac { 5060597}{967680}}\,{n}^{3}{a}^{3}+{\frac {5441}{4032}}\,{n}^{2}{a}^{3}- {\frac {40657}{3840}}\,{n}^{2}{a}^{4}-{\frac {5335139}{725760}}\,{n}^{2 }{a}^{5}+{\frac {310411}{12096}}\,{n}^{2}{a}^{2}+{\frac {14873}{1120}} \,n{a}^{2}+{\frac {11661}{2240}}\,n{a}^{3}+{\frac {3047}{960}}\,n{a}^{4 }-{\frac {226349}{40320}}\,{n}^{2}a+{\frac {50}{3}}\,na-{\frac {7414199 }{580608}}\,{n}^{4}a-{\frac {6936221}{1161216}}\,{n}^{5}a-{\frac { 16466053}{1451520}}\,{n}^{4}{a}^{2}-{\frac {11157193}{1161216}}\,{n}^{5 }{a}^{2}-{\frac {289837}{20736}}\,{n}^{4}{a}^{3}-{\frac {5222461}{ 193536}}\,{n}^{3}{a}^{4}-{\frac {41554217}{2903040}}\,{n}^{4}{a}^{4}-{ \frac {3587317}{322560}}\,{n}^{3}{a}^{5}-{\frac {202389}{17920}}\,{n}^{ 2}{a}^{6}-{\frac {8275991}{967680}}\,{n}^{3}{a}^{6}-{\frac {33945}{7168 }}\,{n}^{2}{a}^{7}-{\frac {144857}{90720}}\,n{a}^{8}-{\frac {1419683}{ 645120}}\,{n}^{2}{a}^{8}-{\frac {1278173}{1935360}}\,n{a}^{9}-{\frac { 263161}{1161216}}\,n{a}^{10}+{\frac {1595}{24}}\,{n}^{2}+{\frac { 1056967}{13440}}\,{n}^{3}+{\frac {972641}{53760}}\,{n}^{4}+{\frac {49}{ 12}}\,n-{\frac {105845}{110592}}\,{n}^{7}+{\frac {5189}{107520}}\,{n}^{ 5}-{\frac {1377331}{1161216}}\,{n}^{6} Theorem Number, 6, : the , 6, -th factorial moment of X is 15 14 2 14 13 2 12 3 38627 a n 28003 a n 147061 a n 1277389 a n 290783 a n ----------- + ------------ + ------------ + -------------- + ------------- 79073280 150958080 18450432 150958080 118609920 12 2 3 11 4 10 3 10 5725211 a n 146099 n a 337753 n a 286779469 n a + -------------- + ------------- + ------------- + ---------------- 42577920 1774080 21288960 276756480 4 9 5 8 4 8 6 6 7033175 n a 713579 n a 75857801 n a 2334385 n a + ------------- + ------------ + -------------- + ------------- 15095808 10063872 16773120 10063872 5 6 5 7 7 4 6 4 617146471 n a 15620635 n a 82825 n a 149079479 n a + --------------- + -------------- + ----------- + --------------- 55351296 10063872 192192 10063872 6 5 8 7 8 2 319795093 n a 82825 n a 159898225 n a 82825 n a + --------------- + ---------- + -------------- + ----------- 110702592 96096 18450432 192192 7 2 7 3 16 6 16137375 n a 162379325 n a 18323 a n 4855735373 n a + -------------- + --------------- + ----------- + --------------- 2050048 55351296 1660538880 110702592 6 2 6 3 5 3 3820697161 n a 3950124985 n a 47630949503 n a + ---------------- + ---------------- + ----------------- 110702592 166053888 553512960 5 4 5 5 4 5 6072307433 n a 5325561785 n a 5478581719 n a + ---------------- + ---------------- + ---------------- 103783680 166053888 63866880 4 6 4 7 3 7 13166161637 n a 4048761103 n a 12545121503 n a + ----------------- + ---------------- + ----------------- 276756480 207567360 276756480 3 8 3 9 2 9 31362635911 n a 461658823 n a 2106350999 n a + ----------------- + --------------- + ---------------- 1660538880 75479040 184504320 2 10 2 11 11 12 565424273 n a 798243007 n a 163287907 n a 83518277 n a + ---------------- + ---------------- + --------------- + -------------- 150958080 830269440 150958080 276756480 13 1352867 n a / 16385 6 565 6 2 565 6 3 + ------------- + |-120 - ----- n a - --- n a - ---- n a 21288960 \ 2048 128 2048 11365 5 3 8475 5 4 339 5 5 115415 4 5 2605 4 6 - ----- n a - ---- n a - ---- n a - ------ n a - ---- n a 512 2048 1024 6144 1024 1695 4 7 44725 3 7 11835 3 8 295 3 9 2324201 2 9 - ----- n a - ----- n a - ----- n a - ----- n a - ------- n a 14336 5376 14336 10752 1161216 1661 2 10 4417799 10 277117 11 9261521 12 2165 2 11 - ----- n a - ------- a - ------ a - -------- a - ------ n a 10752 1451520 362880 63866880 473088 863725 11 208433 12 1073 13 45121 7 329443 8 - ------- n a - -------- n a - ------- n a - ----- a - ------ a 3193344 12773376 3075072 840 13440 334273 9 1529891 5 2367713 6 10468421 7 61284983 3 - ------ a - ------- n a - ------- n a - -------- n a - -------- n a 34560 4480 13440 120960 145152 78479623 3 2 30085751 3 3 7047961 2 3 215217221 2 4 - -------- n a - -------- n a - ------- n a - --------- n a 145152 72576 10080 483840 143956843 2 5 122092031 2 2 912872761 2 906813 3 - --------- n a - --------- n a - --------- n a - ------ n a 483840 147840 1441440 1540 22050169 4 2 3 13399873 2 31876727 - -------- n a - 548/3 a - 1189/6 a - -------- n a - -------- n a 49280 36960 144144 59286085 4 144025 5 62273525 4 2 151415 5 2 - -------- n a - ------ n a - -------- n a - ------ n a 290304 3072 290304 3072 5471515 4 3 4381429 3 4 384745 4 4 4678061 3 5 - ------- n a - ------- n a - ------ n a - ------- n a 41472 18144 6144 41472 6073841 2 6 1620515 3 6 4452449 2 7 30569327 8 - ------- n a - ------- n a - ------- n a - -------- n a 48384 41472 96768 967680 4948831 2 8 26877281 9 6071923 10 465680807 2 - ------- n a - -------- n a - ------- n a - --------- n 387072 2903040 2903040 576576 1507520759 3 85537511 4 16523 5 190297 6 4411 4 - ---------- n - -------- n - ----- a - ------ a - ---- a - 274/3 a 2661120 290304 112 2016 24 14009 3955 7 58320691 5 108379 6 18323 15 - ----- n - ---- n - -------- n - ------ n - --------- a 24 2048 580608 6144 830269440 5 6 82949 14 1083331 13\ 181220821 n a 3791569 n a - -------- a - -------- a | E - -------------- + ------------ 83026944 63866880 / 34594560 329472 7 3 3 2 3 3 226962481 n a 16843834009 n a 2689660447 n a 47203055 n a + -------------- + ---------------- - ---------------- - -------------- 17297280 69189120 14826240 3953664 2 3 2 4 2 5 2 2 49922501 n a 54786941 n a 3262597759 n a 206467531 n a - -------------- + -------------- + ---------------- - --------------- 960960 3144960 103783680 1153152 2 3 4 2 21228089 n a 12798299 n a 44556637 n a 502679 n a 274 n a - ------------- - ------------- - ------------- - ----------- - ------- 288288 320320 1441440 1153152 3 4 5 4 2 5 2 6900660563 n a 4635422833 n a 744707419 n a 1077903049 n a + --------------- + --------------- - --------------- + ---------------- 83026944 166053888 415134720 13837824 4 3 3 4 4 4 11601842723 n a 2126160347 n a 20275451093 n a + ----------------- + ---------------- + ----------------- 138378240 13837824 138378240 3 5 2 6 3 6 212620449 n a 1009807009 n a 13433951023 n a + --------------- + ---------------- + ----------------- 2562560 13837824 138378240 2 7 8 2 8 9 1162234141 n a 599070499 n a 844809127 n a 252177547 n a + ---------------- + -------------- + --------------- + -------------- 27675648 51891840 30750720 39536640 10 2 3 4 8 1260889639 n a 34903 n 590194379 n 74652729 n 230218475 n + ---------------- - -------- - ------------ - ----------- + ------------ 415134720 96 1153152 366080 110702592 9 7 5 6 82825 n 11153 n 2791723679 n 2113281173 n 805412089 n + -------- - ------- + ------------- - ------------- - ------------ 576576 504 332107776 46126080 118609920 and in Maple format 38627/79073280*a^15*n+28003/150958080*a^14*n^2+147061/18450432*a^14*n+1277389/ 150958080*a^13*n^2+290783/118609920*a^12*n^3+5725211/42577920*a^12*n^2+146099/ 1774080*n^3*a^11+337753/21288960*n^4*a^10+286779469/276756480*n^3*a^10+7033175/ 15095808*n^4*a^9+713579/10063872*n^5*a^8+75857801/16773120*n^4*a^8+2334385/ 10063872*n^6*a^6+617146471/55351296*n^5*a^6+15620635/10063872*n^5*a^7+82825/ 192192*n^7*a^4+149079479/10063872*n^6*a^4+319795093/110702592*n^6*a^5+82825/ 96096*n^8*a+159898225/18450432*n^7*a+82825/192192*n^8*a^2+16137375/2050048*n^7* a^2+162379325/55351296*n^7*a^3+18323/1660538880*a^16*n+4855735373/110702592*n^6 *a+3820697161/110702592*n^6*a^2+3950124985/166053888*n^6*a^3+47630949503/ 553512960*n^5*a^3+6072307433/103783680*n^5*a^4+5325561785/166053888*n^5*a^5+ 5478581719/63866880*n^4*a^5+13166161637/276756480*n^4*a^6+4048761103/207567360* n^4*a^7+12545121503/276756480*n^3*a^7+31362635911/1660538880*n^3*a^8+461658823/ 75479040*n^3*a^9+2106350999/184504320*n^2*a^9+565424273/150958080*n^2*a^10+ 798243007/830269440*n^2*a^11+163287907/150958080*n*a^11+83518277/276756480*n*a^ 12+1352867/21288960*n*a^13+(-120-16385/2048*n^6*a-565/128*n^6*a^2-565/2048*n^6* a^3-11365/512*n^5*a^3-8475/2048*n^5*a^4-339/1024*n^5*a^5-115415/6144*n^4*a^5-\ 2605/1024*n^4*a^6-1695/14336*n^4*a^7-44725/5376*n^3*a^7-11835/14336*n^3*a^8-295 /10752*n^3*a^9-2324201/1161216*n^2*a^9-1661/10752*n^2*a^10-4417799/1451520*a^10 -277117/362880*a^11-9261521/63866880*a^12-2165/473088*n^2*a^11-863725/3193344*n *a^11-208433/12773376*n*a^12-1073/3075072*n*a^13-45121/840*a^7-329443/13440*a^8 -334273/34560*a^9-1529891/4480*n*a^5-2367713/13440*n*a^6-10468421/120960*n*a^7-\ 61284983/145152*n^3*a-78479623/145152*n^3*a^2-30085751/72576*n^3*a^3-7047961/ 10080*n^2*a^3-215217221/483840*n^2*a^4-143956843/483840*n^2*a^5-122092031/ 147840*n^2*a^2-912872761/1441440*n*a^2-906813/1540*n*a^3-22050169/49280*n*a^4-\ 548/3*a^2-1189/6*a^3-13399873/36960*n^2*a-31876727/144144*n*a-59286085/290304*n ^4*a-144025/3072*n^5*a-62273525/290304*n^4*a^2-151415/3072*n^5*a^2-5471515/ 41472*n^4*a^3-4381429/18144*n^3*a^4-384745/6144*n^4*a^4-4678061/41472*n^3*a^5-\ 6073841/48384*n^2*a^6-1620515/41472*n^3*a^6-4452449/96768*n^2*a^7-30569327/ 967680*n*a^8-4948831/387072*n^2*a^8-26877281/2903040*n*a^9-6071923/2903040*n*a^ 10-465680807/576576*n^2-1507520759/2661120*n^3-85537511/290304*n^4-16523/112*a^ 5-190297/2016*a^6-4411/24*a^4-274/3*a-14009/24*n-3955/2048*n^7-58320691/580608* n^5-108379/6144*n^6-18323/830269440*a^15-82949/83026944*a^14-1083331/63866880*a ^13)*E-181220821/34594560*n*a^5+3791569/329472*n*a^6+226962481/17297280*n*a^7+ 16843834009/69189120*n^3*a-2689660447/14826240*n^3*a^2-47203055/3953664*n^3*a^3 -49922501/960960*n^2*a^3+54786941/3144960*n^2*a^4+3262597759/103783680*n^2*a^5-\ 206467531/1153152*n^2*a^2-21228089/288288*n*a^2-12798299/320320*n*a^3-44556637/ 1441440*n*a^4-502679/1153152*n^2*a-274/3*n*a+6900660563/83026944*n^4*a+ 4635422833/166053888*n^5*a-744707419/415134720*n^4*a^2+1077903049/13837824*n^5* a^2+11601842723/138378240*n^4*a^3+2126160347/13837824*n^3*a^4+20275451093/ 138378240*n^4*a^4+212620449/2562560*n^3*a^5+1009807009/13837824*n^2*a^6+ 13433951023/138378240*n^3*a^6+1162234141/27675648*n^2*a^7+599070499/51891840*n* a^8+844809127/30750720*n^2*a^8+252177547/39536640*n*a^9+1260889639/415134720*n* a^10-34903/96*n^2-590194379/1153152*n^3-74652729/366080*n^4+230218475/110702592 *n^8+82825/576576*n^9-11153/504*n+2791723679/332107776*n^7-2113281173/46126080* n^5-805412089/118609920*n^6 and in latex {\frac {38627}{79073280}}\,{a}^{15}n+{\frac {28003}{150958080}}\,{a}^{ 14}{n}^{2}+{\frac {147061}{18450432}}\,{a}^{14}n+{\frac {1277389}{ 150958080}}\,{a}^{13}{n}^{2}+{\frac {290783}{118609920}}\,{a}^{12}{n}^{ 3}+{\frac {5725211}{42577920}}\,{a}^{12}{n}^{2}+{\frac {146099}{1774080 }}\,{n}^{3}{a}^{11}+{\frac {337753}{21288960}}\,{n}^{4}{a}^{10}+{\frac {286779469}{276756480}}\,{n}^{3}{a}^{10}+{\frac {7033175}{15095808}}\,{ n}^{4}{a}^{9}+{\frac {713579}{10063872}}\,{n}^{5}{a}^{8}+{\frac { 75857801}{16773120}}\,{n}^{4}{a}^{8}+{\frac {2334385}{10063872}}\,{n}^{ 6}{a}^{6}+{\frac {617146471}{55351296}}\,{n}^{5}{a}^{6}+{\frac { 15620635}{10063872}}\,{n}^{5}{a}^{7}+{\frac {82825}{192192}}\,{n}^{7}{a }^{4}+{\frac {149079479}{10063872}}\,{n}^{6}{a}^{4}+{\frac {319795093}{ 110702592}}\,{n}^{6}{a}^{5}+{\frac {82825}{96096}}\,{n}^{8}a+{\frac { 159898225}{18450432}}\,{n}^{7}a+{\frac {82825}{192192}}\,{n}^{8}{a}^{2} +{\frac {16137375}{2050048}}\,{n}^{7}{a}^{2}+{\frac {162379325}{ 55351296}}\,{n}^{7}{a}^{3}+{\frac {18323}{1660538880}}\,{a}^{16}n+{ \frac {4855735373}{110702592}}\,{n}^{6}a+{\frac {3820697161}{110702592} }\,{n}^{6}{a}^{2}+{\frac {3950124985}{166053888}}\,{n}^{6}{a}^{3}+{ \frac {47630949503}{553512960}}\,{n}^{5}{a}^{3}+{\frac {6072307433}{ 103783680}}\,{n}^{5}{a}^{4}+{\frac {5325561785}{166053888}}\,{n}^{5}{a} ^{5}+{\frac {5478581719}{63866880}}\,{n}^{4}{a}^{5}+{\frac {13166161637 }{276756480}}\,{n}^{4}{a}^{6}+{\frac {4048761103}{207567360}}\,{n}^{4}{ a}^{7}+{\frac {12545121503}{276756480}}\,{n}^{3}{a}^{7}+{\frac { 31362635911}{1660538880}}\,{n}^{3}{a}^{8}+{\frac {461658823}{75479040}} \,{n}^{3}{a}^{9}+{\frac {2106350999}{184504320}}\,{n}^{2}{a}^{9}+{ \frac {565424273}{150958080}}\,{n}^{2}{a}^{10}+{\frac {798243007}{ 830269440}}\,{n}^{2}{a}^{11}+{\frac {163287907}{150958080}}\,n{a}^{11}+ {\frac {83518277}{276756480}}\,n{a}^{12}+{\frac {1352867}{21288960}}\,n {a}^{13}+ \left( -120-{\frac {16385}{2048}}\,{n}^{6}a-{\frac {565}{128} }\,{n}^{6}{a}^{2}-{\frac {565}{2048}}\,{n}^{6}{a}^{3}-{\frac {11365}{ 512}}\,{n}^{5}{a}^{3}-{\frac {8475}{2048}}\,{n}^{5}{a}^{4}-{\frac {339} {1024}}\,{n}^{5}{a}^{5}-{\frac {115415}{6144}}\,{n}^{4}{a}^{5}-{\frac { 2605}{1024}}\,{n}^{4}{a}^{6}-{\frac {1695}{14336}}\,{n}^{4}{a}^{7}-{ \frac {44725}{5376}}\,{n}^{3}{a}^{7}-{\frac {11835}{14336}}\,{n}^{3}{a} ^{8}-{\frac {295}{10752}}\,{n}^{3}{a}^{9}-{\frac {2324201}{1161216}}\,{ n}^{2}{a}^{9}-{\frac {1661}{10752}}\,{n}^{2}{a}^{10}-{\frac {4417799}{ 1451520}}\,{a}^{10}-{\frac {277117}{362880}}\,{a}^{11}-{\frac {9261521} {63866880}}\,{a}^{12}-{\frac {2165}{473088}}\,{n}^{2}{a}^{11}-{\frac { 863725}{3193344}}\,n{a}^{11}-{\frac {208433}{12773376}}\,n{a}^{12}-{ \frac {1073}{3075072}}\,n{a}^{13}-{\frac {45121}{840}}\,{a}^{7}-{\frac {329443}{13440}}\,{a}^{8}-{\frac {334273}{34560}}\,{a}^{9}-{\frac { 1529891}{4480}}\,n{a}^{5}-{\frac {2367713}{13440}}\,n{a}^{6}-{\frac { 10468421}{120960}}\,n{a}^{7}-{\frac {61284983}{145152}}\,{n}^{3}a-{ \frac {78479623}{145152}}\,{n}^{3}{a}^{2}-{\frac {30085751}{72576}}\,{n }^{3}{a}^{3}-{\frac {7047961}{10080}}\,{n}^{2}{a}^{3}-{\frac {215217221 }{483840}}\,{n}^{2}{a}^{4}-{\frac {143956843}{483840}}\,{n}^{2}{a}^{5}- {\frac {122092031}{147840}}\,{n}^{2}{a}^{2}-{\frac {912872761}{1441440} }\,n{a}^{2}-{\frac {906813}{1540}}\,n{a}^{3}-{\frac {22050169}{49280}} \,n{a}^{4}-{\frac {548}{3}}\,{a}^{2}-{\frac {1189}{6}}\,{a}^{3}-{\frac {13399873}{36960}}\,{n}^{2}a-{\frac {31876727}{144144}}\,na-{\frac { 59286085}{290304}}\,{n}^{4}a-{\frac {144025}{3072}}\,{n}^{5}a-{\frac { 62273525}{290304}}\,{n}^{4}{a}^{2}-{\frac {151415}{3072}}\,{n}^{5}{a}^{ 2}-{\frac {5471515}{41472}}\,{n}^{4}{a}^{3}-{\frac {4381429}{18144}}\,{ n}^{3}{a}^{4}-{\frac {384745}{6144}}\,{n}^{4}{a}^{4}-{\frac {4678061}{ 41472}}\,{n}^{3}{a}^{5}-{\frac {6073841}{48384}}\,{n}^{2}{a}^{6}-{ \frac {1620515}{41472}}\,{n}^{3}{a}^{6}-{\frac {4452449}{96768}}\,{n}^{ 2}{a}^{7}-{\frac {30569327}{967680}}\,n{a}^{8}-{\frac {4948831}{387072} }\,{n}^{2}{a}^{8}-{\frac {26877281}{2903040}}\,n{a}^{9}-{\frac {6071923 }{2903040}}\,n{a}^{10}-{\frac {465680807}{576576}}\,{n}^{2}-{\frac { 1507520759}{2661120}}\,{n}^{3}-{\frac {85537511}{290304}}\,{n}^{4}-{ \frac {16523}{112}}\,{a}^{5}-{\frac {190297}{2016}}\,{a}^{6}-{\frac { 4411}{24}}\,{a}^{4}-{\frac {274}{3}}\,a-{\frac {14009}{24}}\,n-{\frac { 3955}{2048}}\,{n}^{7}-{\frac {58320691}{580608}}\,{n}^{5}-{\frac { 108379}{6144}}\,{n}^{6}-{\frac {18323}{830269440}}\,{a}^{15}-{\frac { 82949}{83026944}}\,{a}^{14}-{\frac {1083331}{63866880}}\,{a}^{13} \right) E-{\frac {181220821}{34594560}}\,n{a}^{5}+{\frac {3791569}{ 329472}}\,n{a}^{6}+{\frac {226962481}{17297280}}\,n{a}^{7}+{\frac { 16843834009}{69189120}}\,{n}^{3}a-{\frac {2689660447}{14826240}}\,{n}^{ 3}{a}^{2}-{\frac {47203055}{3953664}}\,{n}^{3}{a}^{3}-{\frac {49922501} {960960}}\,{n}^{2}{a}^{3}+{\frac {54786941}{3144960}}\,{n}^{2}{a}^{4}+{ \frac {3262597759}{103783680}}\,{n}^{2}{a}^{5}-{\frac {206467531}{ 1153152}}\,{n}^{2}{a}^{2}-{\frac {21228089}{288288}}\,n{a}^{2}-{\frac { 12798299}{320320}}\,n{a}^{3}-{\frac {44556637}{1441440}}\,n{a}^{4}-{ \frac {502679}{1153152}}\,{n}^{2}a-{\frac {274}{3}}\,na+{\frac { 6900660563}{83026944}}\,{n}^{4}a+{\frac {4635422833}{166053888}}\,{n}^{ 5}a-{\frac {744707419}{415134720}}\,{n}^{4}{a}^{2}+{\frac {1077903049}{ 13837824}}\,{n}^{5}{a}^{2}+{\frac {11601842723}{138378240}}\,{n}^{4}{a} ^{3}+{\frac {2126160347}{13837824}}\,{n}^{3}{a}^{4}+{\frac {20275451093 }{138378240}}\,{n}^{4}{a}^{4}+{\frac {212620449}{2562560}}\,{n}^{3}{a}^ {5}+{\frac {1009807009}{13837824}}\,{n}^{2}{a}^{6}+{\frac {13433951023} {138378240}}\,{n}^{3}{a}^{6}+{\frac {1162234141}{27675648}}\,{n}^{2}{a} ^{7}+{\frac {599070499}{51891840}}\,n{a}^{8}+{\frac {844809127}{ 30750720}}\,{n}^{2}{a}^{8}+{\frac {252177547}{39536640}}\,n{a}^{9}+{ \frac {1260889639}{415134720}}\,n{a}^{10}-{\frac {34903}{96}}\,{n}^{2}- {\frac {590194379}{1153152}}\,{n}^{3}-{\frac {74652729}{366080}}\,{n}^{ 4}+{\frac {230218475}{110702592}}\,{n}^{8}+{\frac {82825}{576576}}\,{n} ^{9}-{\frac {11153}{504}}\,n+{\frac {2791723679}{332107776}}\,{n}^{7}-{ \frac {2113281173}{46126080}}\,{n}^{5}-{\frac {805412089}{118609920}}\, {n}^{6} Theorem Number, 7, : the , 7, -th factorial moment of X is 19 18 17 2 17 124799 a n 7945759 a n 748961 a n 196757227 a n - ------------ - ------------- - ------------- - --------------- 239117598720 239117598720 59779399680 239117598720 16 2 15 3 15 2 14 3 189463283 a n 25981733 a n 282343109 a n 657932263 a n - ---------------- - --------------- - ---------------- - ---------------- 239117598720 119558799360 14944849920 59779399680 13 4 13 3 12 4 11 5 20527753 a n 3232298371 a n 389727031 a n 307589 a n - --------------- - ----------------- - ---------------- - ------------- 9963233280 14944849920 4269957120 21897216 11 4 10 5 6 9 5 9 1260587855 a n 2775050717 a n 3321715 n a 9587729617 n a - ----------------- - ----------------- - ------------- - ---------------- 853991424 5693276160 51757056 1552711680 6 8 7 7 6 7 8 5 6401399 n a 13069555 n a 83275109 n a 3612685 n a - ------------- - -------------- - -------------- - ------------- 3833856 80510976 5271552 15814656 7 5 7 6 9 3 8 3 1097316089 n a 695339105 n a 33965825 n a 1755757225 n a - ---------------- - --------------- - -------------- - ---------------- 47443968 189775872 253034496 94887936 8 4 9 9 2 2190005 n a 1460530475 n a 781213975 n a / - ------------- - --------------- - --------------- + |720 479232 253034496 253034496 \ 124799 18 1345093 17 8539691 16 591389 15 + ------------ a + ----------- a + ---------- a + --------- a n 119558799360 19926466560 4981616640 383201280 779 14 2 17102989 14 12293 13 2 275 12 3 + ------- a n + --------- a n + ------ a n + ----- a n 1892352 452874240 585728 73728 9204575 12 2 22615 3 11 405 4 10 16997 3 10 + -------- a n + ------ n a + ----- n a + ----- n a 21897216 135168 16384 6144 63005 4 9 1743 5 8 90135 4 8 2065 6 6 108017 5 6 + ----- n a + ----- n a + ----- n a + ---- n a + ------ n a 73728 16384 8192 8192 4096 11249 5 7 3935 7 4 850955 6 4 68677 6 5 19675 8 + ----- n a + ---- n a + ------ n a + ----- n a + ----- n a 4096 8192 24576 12288 16384 130555 7 19675 8 2 23715 7 2 22345 7 3 2363 16 + ------ n a + ----- n a + ----- n a + ----- n a + -------- a n 4096 32768 1024 4096 98402304 2690681 6 5120969 6 2 1903699 6 3 162635453 5 3 + ------- n a + ------- n a + ------- n a + --------- n a 12288 24576 18432 248832 1900703 5 4 490189 5 5 660945431 4 5 496800115 4 6 + ------- n a + ------ n a + --------- n a + --------- n a 6144 4096 995328 1990656 211775 4 7 622296749 3 7 106581883 3 8 67229153 3 9 + ------ n a + --------- n a + --------- n a + -------- n a 3072 1741824 995328 2985984 3609891553 2 9 255262343 2 10 29997421 10 10062091 11 + ---------- n a + --------- n a + -------- a + -------- a 34836480 9953280 622080 622080 185570651 12 23613175 2 11 3960023 11 44815099 12 + --------- a + -------- n a + ------- n a + -------- n a 41057280 5474304 249480 13685760 329298103 13 68284 7 6432575 8 12786179 9 15439647601 5 + --------- n a + ----- a + ------- a + -------- a + ----------- n a 711659520 135 24192 103680 4942080 30048707069 6 213464407 7 177295731677 3 + ----------- n a + --------- n a + ------------ n a 15966720 199584 47900160 522755033599 3 2 24386150719 3 3 29410282697 2 3 + ------------ n a + ----------- n a + ----------- n a 95800320 5132160 4717440 37204538953 2 4 16059241849 2 5 18180190541 2 2 + ----------- n a + ----------- n a + 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3225492637}{1153152}}\,{n}^{2}a+{\frac {170604607}{108108}}\,na+{\frac {24325433}{10368}}\,{n}^{4}a+{\frac {401839153}{497664}}\,{n}^{5}a+{ \frac {116210905}{41472}}\,{n}^{4}{a}^{2}+{\frac {990013249}{995328}}\, {n}^{5}{a}^{2}+{\frac {137679889}{62208}}\,{n}^{4}{a}^{3}+{\frac { 207289757}{62208}}\,{n}^{3}{a}^{4}+{\frac {697522895}{497664}}\,{n}^{4} {a}^{4}+{\frac {1265889257}{622080}}\,{n}^{3}{a}^{5}+{\frac {4565149933 }{2488320}}\,{n}^{2}{a}^{6}+{\frac {473580991}{497664}}\,{n}^{3}{a}^{6} +{\frac {645277453}{725760}}\,{n}^{2}{a}^{7}+{\frac {1402881437}{ 2903040}}\,n{a}^{8}+{\frac {1982637527}{5806080}}\,{n}^{2}{a}^{8}+{ \frac {1982191495}{10450944}}\,n{a}^{9}+{\frac {237932971}{3870720}}\,n {a}^{10}+{\frac {29574223691}{5189184}}\,{n}^{2}+{\frac {478096743911}{ 103783680}}\,{n}^{3}+{\frac {403693486963}{143700480}}\,{n}^{4}+{\frac {61975}{8192}}\,{n}^{8}+{\frac {19675}{98304}}\,{n}^{9}+{\frac {13697}{ 12}}\,{a}^{5}+{\frac {7250}{9}}\,{a}^{6}+{\frac {4004}{3}}\,{a}^{4}+588 \,a+{\frac {267643}{72}}\,n+{\frac {2382821}{36864}}\,{n}^{7}+{\frac { 611150509}{497664}}\,{n}^{5}+{\frac {124360691}{373248}}\,{n}^{6}+{ \frac {3535111}{155675520}}\,{a}^{15}+{\frac {1813576621}{9963233280}} \,{a}^{14}+{\frac {78331777}{76640256}}\,{a}^{13} \right) E-{\frac { 3143065657}{39852933120}}\,{a}^{15}n-{\frac {13246420337}{59779399680}} \,{a}^{14}{n}^{2}-{\frac {72979289}{177914880}}\,{a}^{14}n-{\frac { 17477542417}{11955879936}}\,{a}^{13}{n}^{2}-{\frac {908981027}{ 426995712}}\,{a}^{12}{n}^{3}-{\frac {9234193213}{1423319040}}\,{a}^{12} {n}^{2}-{\frac {1401469071143}{119558799360}}\,{n}^{3}{a}^{11}-{\frac { 10898020877}{948879360}}\,{n}^{4}{a}^{10}-{\frac {9844765505}{229920768 }}\,{n}^{3}{a}^{10}-{\frac {1499661161279}{29889699840}}\,{n}^{4}{a}^{9 }-{\frac {15674976241}{426995712}}\,{n}^{5}{a}^{8}-{\frac {224205021421 }{1494484992}}\,{n}^{4}{a}^{8}-{\frac {116825148665}{1707982848}}\,{n}^ {6}{a}^{6}-{\frac {19758259043}{67931136}}\,{n}^{5}{a}^{6}-{\frac { 2877155038061}{23911759872}}\,{n}^{5}{a}^{7}-{\frac {10013653831}{ 142331904}}\,{n}^{7}{a}^{4}-{\frac {23839102751}{77635584}}\,{n}^{6}{a} ^{4}-{\frac {1342579522303}{8539914240}}\,{n}^{6}{a}^{5}-{\frac { 3681882635}{142331904}}\,{n}^{8}a-{\frac {120988734079}{853991424}}\,{n }^{7}a-{\frac {18287270065}{569327616}}\,{n}^{8}{a}^{2}-{\frac { 56708281387}{426995712}}\,{n}^{7}{a}^{2}-{\frac {353623169039}{ 3415965696}}\,{n}^{7}{a}^{3}-{\frac {312773963}{29889699840}}\,{a}^{16} n-{\frac {237760775}{253034496}}\,{n}^{10}-{\frac {394108317011}{ 1067489280}}\,{n}^{6}a-{\frac {302456842873}{853991424}}\,{n}^{6}{a}^{2 }-{\frac {389255534483}{1067489280}}\,{n}^{6}{a}^{3}-{\frac { 263901538561}{355829760}}\,{n}^{5}{a}^{3}-{\frac {154933803725}{ 213497856}}\,{n}^{5}{a}^{4}-{\frac {4636240841471}{8539914240}}\,{n}^{5 }{a}^{5}-{\frac {2986080041}{3594240}}\,{n}^{4}{a}^{5}-{\frac { 1699983127553}{2490808320}}\,{n}^{4}{a}^{6}-{\frac {5486238255031}{ 14944849920}}\,{n}^{4}{a}^{7}-{\frac {80366115391}{158146560}}\,{n}^{3} {a}^{7}-{\frac {165100442957}{553512960}}\,{n}^{3}{a}^{8}-{\frac { 1945973249747}{14944849920}}\,{n}^{3}{a}^{9}-{\frac {705544194283}{ 4981616640}}\,{n}^{2}{a}^{9}-{\frac {1910835130993}{29889699840}}\,{n}^ {2}{a}^{10}-{\frac {346078956019}{14944849920}}\,{n}^{2}{a}^{11}-{ \frac {431324188231}{29889699840}}\,n{a}^{11}-{\frac {40704699481}{ 7472424960}}\,n{a}^{12}-{\frac {14314292023}{8539914240}}\,n{a}^{13}+{ \frac {495642881}{5189184}}\,n{a}^{5}-{\frac {531040241}{15966720}}\,n{ a}^{6}-{\frac {13547126867}{177914880}}\,n{a}^{7}-{\frac {104803108433} {69189120}}\,{n}^{3}a+{\frac {2299559028761}{1245404160}}\,{n}^{3}{a}^{ 2}+{\frac {225600057695}{498161664}}\,{n}^{3}{a}^{3}+{\frac { 128022943991}{207567360}}\,{n}^{2}{a}^{3}+{\frac {610861903}{3538080}} \,{n}^{2}{a}^{4}-{\frac {7074724781}{88957440}}\,{n}^{2}{a}^{5}+{\frac {12718125193}{9434880}}\,{n}^{2}{a}^{2}+{\frac {207074213}{432432}}\,n{ a}^{2}+{\frac {682510487}{2162160}}\,n{a}^{3}+{\frac {7101856859}{ 25945920}}\,n{a}^{4}+{\frac {1703771969}{10378368}}\,{n}^{2}a+588\,na-{ \frac {1039055972377}{1868106240}}\,{n}^{4}a-{\frac {134723484719}{ 2134978560}}\,{n}^{5}a+{\frac {4847254935953}{7472424960}}\,{n}^{4}{a}^ {2}-{\frac {3564489950009}{7472424960}}\,{n}^{5}{a}^{2}-{\frac { 1542012318221}{3736212480}}\,{n}^{4}{a}^{3}-{\frac {85813945297}{ 103783680}}\,{n}^{3}{a}^{4}-{\frac {9101603091137}{7472424960}}\,{n}^{4 }{a}^{4}-{\frac {550630065743}{996323328}}\,{n}^{3}{a}^{5}-{\frac { 83132958553}{177914880}}\,{n}^{2}{a}^{6}-{\frac {78959792117}{88957440} }\,{n}^{3}{a}^{6}-{\frac {141328516753}{415134720}}\,{n}^{2}{a}^{7}-{ \frac {135265241}{1621620}}\,n{a}^{8}-{\frac {135603641399}{498161664}} \,{n}^{2}{a}^{8}-{\frac {46629054149}{830269440}}\,n{a}^{9}-{\frac { 3365954797}{104509440}}\,n{a}^{10}+{\frac {74781}{32}}\,{n}^{2}+{\frac {12818384591}{3459456}}\,{n}^{3}+{\frac {404390921533}{207567360}}\,{n} ^{4}-{\frac {20954760755}{747242496}}\,{n}^{8}-{\frac {1458979735}{ 284663808}}\,{n}^{9}+{\frac {3383}{24}}\,n-{\frac {23461009193}{ 711659520}}\,{n}^{7}+{\frac {169376600585}{249080832}}\,{n}^{5}+{\frac {653573779205}{2988969984}}\,{n}^{6} Theorem Number, 8, : the , 8, -th factorial moment of X is / 26487497 20 770951 21 8319151 19 |-5040 - ------------- a - -------------- a - ----------- a \ 7240779786240 18431075819520 64049356800 17373 19 15503081 18 2987 17 2 - ----------- a n - ------------ a n - --------- a n 13243310080 138579517440 104552448 25079897 17 150589 16 2 947 15 3 10982117 15 2 - ---------- a n - -------- a n - ------- a n - --------- a n 6599024640 73801728 2635776 186810624 179011 14 3 17495 13 4 4397347 13 3 3835 12 4 - ------- a n - ------- a n - ------- a n - ----- a n 7907328 5271552 7907328 22528 8455 11 5 125435 11 4 31867 10 5 1613 6 9 - ------ a n - ------ a n - ----- a n - ----- n a 405504 36864 36864 18432 167801 5 9 12067 6 8 3935 7 7 321515 6 7 3935 8 5 - ------ n a - ----- n a - ----- n a - ------ n a - ---- n a 12288 4096 14336 9216 8192 987847 7 5 38065 7 6 19675 9 3 1093925 8 3 3935 8 4 - ------ n a - ----- n a - ----- n a - ------- n a - ---- n a 18432 6144 73728 24576 512 846025 9 452525 9 2 16366451 18 2162700403 17 - ------ n a - ------ n a - ---------- a - ----------- a 73728 73728 6599024640 74724249600 25952611 16 2230996321 15 6699090137 14 2 39781476559 14 - --------- a - ---------- a n - ---------- a n - ----------- a n 113218560 3113510400 7472424960 7925299200 2168981761 13 2 173534917 12 3 9319016111 12 2 - ---------- a n - --------- a n - ---------- a n 266872320 24634368 191600640 647630449 3 11 3842203 4 10 41094109109 3 10 926159743 4 9 - --------- n a - ------- n a - ----------- n a - --------- n a 12317184 110592 156764160 4478976 11908841 5 8 1244433227 4 8 3708005 6 6 2352063311 5 6 - -------- n a - ---------- n a - ------- n a - ---------- n a 110592 1492992 18432 1492992 2269159 5 7 330239 7 4 91579597 6 4 9568573 6 5 - ------- n a - ------ n a - -------- n a - ------- n a 4608 1536 55296 13824 4531075 8 8756105 7 8408995 8 2 5569145 7 2 - ------- n a - ------- n a - ------- n a - ------- n a 36864 9216 73728 6144 762907 7 3 289256053 16 137725 10 2202331033 6 - ------ n a - ---------- a n - ------ n - ---------- n a 1536 4269957120 73728 559872 20718636451 6 2 14134612517 6 3 3031995593 5 3 - ----------- n a - ----------- n a - ---------- n a 4478976 4478976 248832 28597600111 5 4 2986273009 5 5 8332490611 4 5 - ----------- n a - ---------- n a - ---------- n a 3732480 746496 622080 3306877375 4 6 3942020929 4 7 11267711827 3 7 - ---------- n a - ---------- n a - ----------- n a 497664 1492992 1451520 40491794201 3 8 8691058903 3 9 4223670013 2 9 - ----------- n a - ---------- n a - ---------- n a 13063680 8709120 1749600 30232329431 2 10 719777209 10 2836903289 11 9137259721 12 - ----------- n a - --------- a - ---------- a - ---------- a 37324800 1088640 10886400 102643200 25239272789 2 11 241255253441 11 69181054793 12 - ----------- n a - ------------ n a - ----------- n a 114960384 615859200 615859200 98576582263 13 10728481 7 6201227 8 268455443 9 - ----------- n a - -------- a - ------- a - --------- a 3736212480 2160 2160 181440 3034466923 5 2228341297441 6 886104105191 7 - ---------- n a - ------------- n a - ------------ n a 101088 111196800 70761600 13390972149619 3 12417260855933 3 2 7026938912953 3 3 - -------------- n a - -------------- n a - ------------- n a 389188800 222393600 133436160 1039523370187 2 3 5432768574461 2 4 555550861709 2 5 - ------------- n a - ------------- n a - ------------ n a 17690400 111196800 13899600 402701981328833 2 2 33600686069 2 4163808870583 3 - --------------- n a - ----------- n a - ------------- n a 6616209600 933660 110270160 2076360698987 4 2 157061 3 2457998544394 2 - ------------- n a - 8712 a - ------ a - ------------- n a 60147360 15 103378275 78824486749 72622093564103 4 231926196773 5 - ----------- n a - -------------- n a - ------------ n a 6235515 2802159360 20528640 18987676005929 4 2 625194555641 5 2 6621089422501 4 3 - -------------- n a - ------------ n a - ------------- n a 560431872 41057280 215550720 584534104801069 3 4 29116738547 4 4 31699189201577 3 5 - --------------- n a - ----------- n a - -------------- n a 14010796800 1283040 1077753600 748707299471 2 6 249694528381 3 6 6117958065407 2 7 - ------------ n a - ------------ n a - ------------- n a 31449600 15396480 457228800 718199295653 8 176473013327 2 8 1065756100093 9 - ------------ n a - ------------ n a - ------------- n a 111196800 28740096 359251200 456266254631 10 9475171964915 2 938330670333143 3 - ------------ n a - ------------- n - --------------- n 391910400 209513304 23156733600 64607995792937 4 3530941 8 1991075 9 146068 5 1326407 6 - -------------- n - ------- n - ------- n - ------ a - ------- a 2335132800 14336 73728 15 180 325423 4 1431976079 7 1032221065583 5 - ------ a - 4356 a - 245609/9 n - ---------- n - ------------- n 30 1119744 72783360 100713212689 6 13938193567 15 738313842433 14 183661387 13\ - ------------ n - ----------- a - ------------ a - --------- a | E 20528640 10189670400 112086374400 6998400 / 19 18 17 2 62601097961 a n 494454744059 a n 465158599429 a n + ----------------- + ------------------ + ------------------- 53353114214400 37544784076800 14079294028800 17 16 2 15 3 11809237427723 a n 13317520366351 a n 988284826057 a n + -------------------- + --------------------- + ------------------- 119259902361600 39753300787200 2208516710400 15 2 14 3 13 4 6393254283539 a n 655950499908757 a n 4751386719613 a n + -------------------- + ---------------------- + -------------------- 2888060313600 168951528345600 1316505415680 13 3 12 4 11 5 1840652895310519 a n 743962184940097 a n 59192932293307 a n + ----------------------- + ---------------------- + --------------------- 84475764172800 28963119144960 3218124349440 11 4 10 5 121015480643878001 a n 268232667661063 a n + ------------------------- + ---------------------- 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256406305 n a 6283883911579 n a 256406305 n a + --------------- + -------------------- + ---------------- 279351072 429083246592 558702144 10 3 9 4 10 4 924034220755 n a 192933230353 n a 51281261 n a + ------------------- + ------------------ + --------------- 214541623296 5675704320 85954176 9 5 20 2 20 832150943185 n a 137221859 a n 7994784437 a n + ------------------ + ---------------- + ---------------- 143027748864 202741834014720 126713646259200 8 6 16 4 16 3 225229931659 n a 687668339 a n 20342177133001 a n + ------------------ + ---------------- + --------------------- 6285399120 3556874280960 675806113382400 9 6 7 8 8 8 178974430933 n a 3783483694619 n a 3468927935 n a + ------------------ + ------------------- + ---------------- 429083246592 178784686080 19503783936 8 7 6 6 2 22810623881 n a 360133105303438981 n a 878517908387083 n a + ----------------- + ----------------------- + --------------------- 5501067264 126713646259200 281585880576 6 3 5 3 2126258348830741 n a 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----------------------- + ------------------- + --------------------- 21118941043200 83805321600 603398315520 2 3 4 149353819 n 16728508774603 n 22051214606116063 n - ------------ - ----------------- - -------------------- 8640 558702144 1173274502400 11 12 8 1308934438597 n 256406305 n 560670971761673 n + ----------------- + ------------- + ------------------ 429083246592 2234808576 2413593262080 9 7 5 542188324029013 n 744383 n 694407195353689 n 215212198557463 n + ------------------ - -------- - ------------------ - ------------------ 7240779786240 720 2436800889600 26072766720 6 29150235830821579 n - -------------------- 8447576417280 and in Maple format (-5040-26487497/7240779786240*a^20-770951/18431075819520*a^21-8319151/ 64049356800*a^19-17373/13243310080*a^19*n-15503081/138579517440*a^18*n-2987/ 104552448*a^17*n^2-25079897/6599024640*a^17*n-150589/73801728*a^16*n^2-947/ 2635776*a^15*n^3-10982117/186810624*a^15*n^2-179011/7907328*a^14*n^3-17495/ 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614281793139709}{56317176115200}}\,{a}^{14}{n}^{2}+{\frac { 134820521029433}{14481559572480}}\,{a}^{14}n+{\frac {86010026005763}{ 1979900722800}}\,{a}^{13}{n}^{2}+{\frac {93015182160034951}{ 1013709170073600}}\,{a}^{12}{n}^{3}+{\frac {35286260440424963}{ 253427292518400}}\,{a}^{12}{n}^{2}+{\frac {802919472124169}{ 2540624486400}}\,{n}^{3}{a}^{11}+{\frac {42825836612033809}{ 101370917007360}}\,{n}^{4}{a}^{10}+{\frac {216347197901317853}{ 253427292518400}}\,{n}^{3}{a}^{10}+{\frac {18200158175374223}{ 14907487795200}}\,{n}^{4}{a}^{9}+{\frac {77278707331600963}{ 67580611338240}}\,{n}^{5}{a}^{8}+{\frac {43262926076648177}{ 15839205782400}}\,{n}^{4}{a}^{8}+{\frac {218865194949}{120393728}}\,{n} ^{6}{a}^{6}+{\frac {2081447685386653}{438835138560}}\,{n}^{5}{a}^{6}+{ \frac {178864378992926519}{67580611338240}}\,{n}^{5}{a}^{7}+{\frac { 6801944074422641}{4137588449280}}\,{n}^{7}{a}^{4}+{\frac { 74602226298527}{17141997600}}\,{n}^{6}{a}^{4}+{\frac {5326814038183}{ 1698278400}}\,{n}^{6}{a}^{5}+{\frac {3047781057829169}{7240779786240}} \,{n}^{8}a+{\frac {12275254591447339}{7240779786240}}\,{n}^{7}a+{\frac {9247895541457241}{14481559572480}}\,{n}^{8}{a}^{2}+{\frac { 1128374762504477}{724077978624}}\,{n}^{7}{a}^{2}+{\frac { 4711570703516869}{2633010831360}}\,{n}^{7}{a}^{3}+{\frac { 564620983321979}{1013709170073600}}\,{a}^{16}n+{\frac {756211747}{ 50685458503680}}\,{a}^{18}{n}^{3}+{\frac {9680353493}{5081248972800}}\, {a}^{18}{n}^{2}+{\frac {44499487}{779776284672}}\,{a}^{19}{n}^{2}+{ \frac {48042171563}{3975330078720}}\,{a}^{15}{n}^{4}+{\frac {5649583741 }{5335311421440}}\,{a}^{17}{n}^{3}+{\frac {453363820911563}{ 19308746096640}}\,{n}^{10}+{\frac {104321785}{8736989184}}\,{a}^{12}{n} ^{6}+{\frac {17880564553591}{9654373048320}}\,{a}^{12}{n}^{5}+{\frac { 53171377211}{567904296960}}\,{a}^{13}{n}^{5}+{\frac {154760513813}{ 8251600896}}\,{n}^{10}a+{\frac {156829378739}{87767027712}}\,{a}^{9}{n} ^{7}+{\frac {256406305}{279351072}}\,{n}^{11}a+{\frac {6283883911579}{ 429083246592}}\,{n}^{10}{a}^{2}+{\frac {256406305}{558702144}}\,{n}^{11 }{a}^{2}+{\frac {924034220755}{214541623296}}\,{n}^{10}{a}^{3}+{\frac { 192933230353}{5675704320}}\,{n}^{9}{a}^{4}+{\frac {51281261}{85954176}} \,{n}^{10}{a}^{4}+{\frac {832150943185}{143027748864}}\,{n}^{9}{a}^{5}+ {\frac {137221859}{202741834014720}}\,{a}^{20}{n}^{2}+{\frac { 7994784437}{126713646259200}}\,{a}^{20}n+{\frac {225229931659}{ 6285399120}}\,{n}^{8}{a}^{6}+{\frac {687668339}{3556874280960}}\,{a}^{ 16}{n}^{4}+{\frac {20342177133001}{675806113382400}}\,{a}^{16}{n}^{3}+{ \frac {178974430933}{429083246592}}\,{n}^{9}{a}^{6}+{\frac { 3783483694619}{178784686080}}\,{n}^{7}{a}^{8}+{\frac {3468927935}{ 19503783936}}\,{n}^{8}{a}^{8}+{\frac {22810623881}{5501067264}}\,{n}^{8 }{a}^{7}+{\frac {360133105303438981}{126713646259200}}\,{n}^{6}a+{ \frac {878517908387083}{281585880576}}\,{n}^{6}{a}^{2}+{\frac { 2126258348830741}{517198556160}}\,{n}^{6}{a}^{3}+{\frac { 370950768081825311}{63356823129600}}\,{n}^{5}{a}^{3}+{\frac { 1926785235283785673}{253427292518400}}\,{n}^{5}{a}^{4}+{\frac { 136704996863}{275839229952}}\,{a}^{11}{n}^{6}+{\frac { 164638860514446011}{24135932620800}}\,{n}^{5}{a}^{5}+{\frac { 942344605364917033}{126713646259200}}\,{n}^{4}{a}^{5}+{\frac { 15897052816723999}{1979900722800}}\,{n}^{4}{a}^{6}+{\frac { 67486151731083233}{13338278553600}}\,{n}^{4}{a}^{7}+{\frac { 3329761646783609}{635156121600}}\,{n}^{3}{a}^{7}+{\frac { 317834562168976309}{84475764172800}}\,{n}^{3}{a}^{8}+{\frac { 2762518913570987}{1407929402880}}\,{n}^{3}{a}^{9}+{\frac {5640143392417 }{3555377280}}\,{n}^{2}{a}^{9}+{\frac {98287669228487}{114672983040}}\, {n}^{2}{a}^{10}+{\frac {8625523}{4770396094464}}\,{a}^{21}n+{\frac { 770951}{36862151639040}}\,{a}^{22}n+{\frac {558500641}{305794621440}}\, {a}^{14}{n}^{5}+{\frac {4362434181601}{14907487795200}}\,{a}^{14}{n}^{4 }+{\frac {8690028999712897}{23038844774400}}\,{n}^{2}{a}^{11}+{\frac { 885046865359741}{5171985561600}}\,n{a}^{11}+{\frac {463370704566281}{ 6033983155200}}\,n{a}^{12}+{\frac {2100675053291657}{72407797862400}}\, n{a}^{13}+{\frac {2837050789}{53635405824}}\,{a}^{10}{n}^{7}+{\frac { 74983649011331}{9654373048320}}\,{a}^{10}{n}^{6}-{\frac {2642185516853} {2285599680}}\,n{a}^{5}-{\frac {787149763}{8868288}}\,n{a}^{6}+{\frac { 73228378171769}{167610643200}}\,n{a}^{7}+{\frac {9392152744816141}{ 879955876800}}\,{n}^{3}a-{\frac {1011321355120619}{54997242300}}\,{n}^{ 3}{a}^{2}-{\frac {9375282454895887}{1407929402880}}\,{n}^{3}{a}^{3}-{ \frac {550189158445237}{83805321600}}\,{n}^{2}{a}^{3}-{\frac { 863073308678171}{251415964800}}\,{n}^{2}{a}^{4}-{\frac {95936799895753} {150849578880}}\,{n}^{2}{a}^{5}-{\frac {351103792708733}{31426995600}} \,{n}^{2}{a}^{2}-{\frac {19837257809}{5542680}}\,n{a}^{2}-{\frac { 6032441087}{2238390}}\,n{a}^{3}-{\frac {300853692257}{119721888}}\,n{a} ^{4}-{\frac {289684616866}{130945815}}\,{n}^{2}a-4356\,na+{\frac { 41845855564374689}{10559470521600}}\,{n}^{4}a-{\frac {22311287599405757 }{25342729251840}}\,{n}^{5}a-{\frac {66765366149430107}{6335682312960}} \,{n}^{4}{a}^{2}+{\frac {38557404217681}{19615115520}}\,{n}^{5}{a}^{2}+ {\frac {4332793285452323}{4223788208640}}\,{n}^{4}{a}^{3}+{\frac { 917402170289399}{234654900480}}\,{n}^{3}{a}^{4}+{\frac { 3350789086506761}{348114412800}}\,{n}^{4}{a}^{4}+{\frac { 141936425063519861}{42237882086400}}\,{n}^{3}{a}^{5}+{\frac { 72463022937523}{23944377600}}\,{n}^{2}{a}^{6}+{\frac { 110211997881428959}{14079294028800}}\,{n}^{3}{a}^{6}+{\frac { 215925554443}{78140160}}\,{n}^{2}{a}^{7}+{\frac {948656452989299}{ 1508495788800}}\,n{a}^{8}+{\frac {54904776967062751}{21118941043200}}\, {n}^{2}{a}^{8}+{\frac {41841407817563}{83805321600}}\,n{a}^{9}+{\frac { 197910327476921}{603398315520}}\,n{a}^{10}-{\frac {149353819}{8640}}\,{ n}^{2}-{\frac {16728508774603}{558702144}}\,{n}^{3}-{\frac { 22051214606116063}{1173274502400}}\,{n}^{4}+{\frac {1308934438597}{ 429083246592}}\,{n}^{11}+{\frac {256406305}{2234808576}}\,{n}^{12}+{ \frac {560670971761673}{2413593262080}}\,{n}^{8}+{\frac { 542188324029013}{7240779786240}}\,{n}^{9}-{\frac {744383}{720}}\,n-{ \frac {694407195353689}{2436800889600}}\,{n}^{7}-{\frac { 215212198557463}{26072766720}}\,{n}^{5}-{\frac {29150235830821579}{ 8447576417280}}\,{n}^{6} Theorem Number, 9, : the , 9, -th factorial moment of X is 25 24 23 2 5528924461 a n 69181822691 a n 231347572201 a n - ------------------- - ------------------ - ------------------- 7612280061139353600 845808895682150400 7612280061139353600 23 22 2 21 3 149140791409 a n 105764643931 a n 485982516233 a n - ------------------ - ------------------- - ------------------- 39037333646868480 31984370004787200 585560004703027200 21 2 20 3 19 4 11327413142593 a n 9053100061433 a n 3227112927221 a n - --------------------- - -------------------- - -------------------- 76891717789286400 115337576683929600 230675153367859200 19 3 18 4 17 5 7744154054750579 a n 895409862051959 a n 2314321201669 a n - ----------------------- - ---------------------- - -------------------- 2537426687046451200 761228006113935360 13354877300244480 17 4 16 5 15 6 101104406399924503 a n 482323917566443 a n 468042508267 a n - ------------------------- - ---------------------- - ------------------- 2537426687046451200 39037333646868480 304613047664640 15 5 14 6 13 7 8776582621034587 a n 98250418052629 a n 2342890043861 a n - ----------------------- - --------------------- - -------------------- 24876732225945600 1066145666826240 236921259294720 13 6 12 7 78733928110625299 a n 8895204002026547 a n - ------------------------ - ----------------------- 36248952672092160 18124476336046080 11 8 11 7 10 8 191434861073773 a n 1382756940912967 a n 1127003074949 a n - ---------------------- - ----------------------- - -------------------- 4027661408010240 149172644741120 616415887360 9 9 9 8 8 9 111387429097907 a n 6791671450708327 a n 40015430376491 a n - 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26431527990067200}}\,{n}^{5}{a}^{6}-{\frac {104187773442977117997179}{ 2378837519106048000}}\,{n}^{5}{a}^{7}-{\frac {311881075379403335083}{ 11327797710028800}}\,{n}^{7}{a}^{4}-{\frac {32380891367371949841133}{ 594709379776512000}}\,{n}^{6}{a}^{4}-{\frac {115154584016510415538723}{ 2378837519106048000}}\,{n}^{6}{a}^{5}-{\frac {379954927485142859021}{ 67966786260172800}}\,{n}^{8}a-{\frac {1100813529062886623437}{ 59470937977651200}}\,{n}^{7}a-{\frac {52861416350571032213}{ 5663898855014400}}\,{n}^{8}{a}^{2}-{\frac {3746892355356066563729}{ 237883751910604800}}\,{n}^{7}{a}^{2}-{\frac {11773981928956204871009}{ 475767503821209600}}\,{n}^{7}{a}^{3}-{\frac {568043001925925969}{ 37169336236032000}}\,{a}^{16}n-{\frac {243873452286758929}{ 3806140030569676800}}\,{a}^{18}{n}^{3}-{\frac {103068037274161111}{ 211452223920537600}}\,{a}^{18}{n}^{2}-{\frac {976563030218272751}{ 19030700152848384000}}\,{a}^{19}{n}^{2}-{\frac {126456824401671259}{ 16519704993792000}}\,{a}^{15}{n}^{4}-{\frac {5968403829072161}{ 7364822040576000}}\,{a}^{17}{n}^{3}-{\frac {2256746762067861883}{ 6178798750924800}}\,{n}^{10}-{\frac {716281849921346329}{ 27186714504069120}}\,{a}^{12}{n}^{6}-{\frac {50683430050776911}{ 190611980697600}}\,{a}^{12}{n}^{5}-{\frac {12653126773730147171}{ 271867145040691200}}\,{a}^{13}{n}^{5}-{\frac {318667787716092151}{ 549226555637760}}\,{n}^{10}a-{\frac {23149748120933617}{48271865241600} }\,{a}^{9}{n}^{7}-{\frac {50064450999527}{762814660608}}\,{n}^{11}a-{ \frac {295859522358462331}{549226555637760}}\,{n}^{10}{a}^{2}-{\frac { 584145986515219}{8136689713152}}\,{n}^{11}{a}^{2}-{\frac { 1169060319735375293}{3295359333826560}}\,{n}^{10}{a}^{3}-{\frac { 6398462224616804779}{4119199167283200}}\,{n}^{9}{a}^{4}-{\frac { 36968146702850659}{183075518545920}}\,{n}^{10}{a}^{4}-{\frac { 930964060983328051}{1267445897625600}}\,{n}^{9}{a}^{5}-{\frac { 1346655970757501}{380614003056967680}}\,{a}^{20}{n}^{2}-{\frac { 77573865791896099}{4757675038212096000}}\,{a}^{20}n-{\frac { 373394068656410881}{171633298636800}}\,{n}^{8}{a}^{6}-{\frac { 61926392432867741}{86503182512947200}}\,{a}^{16}{n}^{4}-{\frac { 21178029667214094491}{3171783358808064000}}\,{a}^{16}{n}^{3}-{\frac { 13189430775241691}{50697835905024}}\,{n}^{9}{a}^{6}-{\frac { 1427756135672963753}{823839833456640}}\,{n}^{7}{a}^{8}-{\frac { 3044486803135361}{15843073720320}}\,{n}^{8}{a}^{8}-{\frac { 1595202510981488923}{2059599583641600}}\,{n}^{8}{a}^{7}-{\frac { 12035521597500698423741}{594709379776512000}}\,{n}^{6}a-{\frac { 1132478047212339120239}{45746875367424000}}\,{n}^{6}{a}^{2}-{\frac { 219615273278972892479}{5286305598013440}}\,{n}^{6}{a}^{3}-{\frac { 855713013947705504263}{19823645992550400}}\,{n}^{5}{a}^{3}-{\frac { 22411732983557319003851}{297354689888256000}}\,{n}^{5}{a}^{4}-{\frac { 16800353555399131723}{90622381680230400}}\,{a}^{11}{n}^{6}-{\frac { 8500649492783197565923}{108128978141184000}}\,{n}^{5}{a}^{5}-{\frac { 4824360026530001451071}{74338672472064000}}\,{n}^{4}{a}^{5}-{\frac { 52381045725383522219}{583048411545600}}\,{n}^{4}{a}^{6}-{\frac { 997910997524325108997}{15650246836224000}}\,{n}^{4}{a}^{7}-{\frac { 888613444622044220579}{16519704993792000}}\,{n}^{3}{a}^{7}-{\frac { 444544673617815199057}{9911822996275200}}\,{n}^{3}{a}^{8}-{\frac { 34714825431166083761}{1304187236352000}}\,{n}^{3}{a}^{9}-{\frac { 72626814104476919863}{4129926248448000}}\,{n}^{2}{a}^{9}-{\frac { 14064288198785523869}{1304187236352000}}\,{n}^{2}{a}^{10}-{\frac { 3265040796232409}{2114522239205376000}}\,{a}^{21}n-{\frac { 1025695801063}{10485234243993600}}\,{a}^{22}n-{\frac { 2011721883773761381}{380614003056967680}}\,{a}^{14}{n}^{5}-{\frac { 63280753871347716901}{1189418759553024000}}\,{a}^{14}{n}^{4}-{\frac { 4314836198979596023}{795066015744000}}\,{n}^{2}{a}^{11}-{\frac { 4459036295215391}{2234808576000}}\,n{a}^{11}-{\frac {40626767470801019} {40226554368000}}\,n{a}^{12}-{\frac {21504173992338127}{49165788672000} }\,n{a}^{13}-{\frac {144969637284824323}{1647679666913280}}\,{a}^{10}{n }^{7}-{\frac {32111098713488503}{37692018524160}}\,{a}^{10}{n}^{6}+{ \frac {14174264240341}{1064194560}}\,n{a}^{5}+{\frac {37473414515773}{ 10158220800}}\,n{a}^{6}-{\frac {7411945698817397}{3352212864000}}\,n{a} ^{7}-{\frac {386667608529708931}{4588806942720}}\,{n}^{3}a+{\frac { 197169738142800726707}{1032481562112000}}\,{n}^{3}{a}^{2}+{\frac { 177957881356718935919}{2064963124224000}}\,{n}^{3}{a}^{3}+{\frac { 108859867066797883}{1536430896000}}\,{n}^{2}{a}^{3}+{\frac { 861466733386018399}{18437170752000}}\,{n}^{2}{a}^{4}+{\frac { 127708370103431363}{7374868300800}}\,{n}^{2}{a}^{5}+{\frac { 17423190755276393}{170714544000}}\,{n}^{2}{a}^{2}+{\frac { 20120809726451}{665121600}}\,n{a}^{2}+{\frac {13763232972373}{547747200 }}\,n{a}^{3}+{\frac {689892939352157}{27935107200}}\,n{a}^{4}+{\frac { 44178086837677}{1719083520}}\,{n}^{2}a+36528\,na-{\frac { 373465111491524593591}{12389778745344000}}\,{n}^{4}a+{\frac { 1075107341066776031873}{49559114981376000}}\,{n}^{5}a+{\frac { 584500624814439959819}{4129926248448000}}\,{n}^{4}{a}^{2}+{\frac { 1025981094692404842371}{148677344944128000}}\,{n}^{5}{a}^{2}+{\frac { 94139198143088089}{5949473587200}}\,{n}^{4}{a}^{3}-{\frac { 17233808584911411847}{2064963124224000}}\,{n}^{3}{a}^{4}-{\frac { 19196101257220950313}{258120390528000}}\,{n}^{4}{a}^{4}-{\frac { 11807504903230243531}{750895681536000}}\,{n}^{3}{a}^{5}-{\frac { 201207000999282919}{10535526144000}}\,{n}^{2}{a}^{6}-{\frac { 577954275129490561349}{8259852496896000}}\,{n}^{3}{a}^{6}-{\frac { 3391404868239923767}{147497366016000}}\,{n}^{2}{a}^{7}-{\frac { 505576628039141}{101582208000}}\,n{a}^{8}-{\frac {1163363698226661689}{ 45888069427200}}\,{n}^{2}{a}^{8}-{\frac {3764735345527777}{812657664000 }}\,n{a}^{9}-{\frac {6857172432393301}{2011327718400}}\,n{a}^{10}+{ \frac {34741699}{240}}\,{n}^{2}+{\frac {10008611095608101}{37246809600} }\,{n}^{3}+{\frac {65872939745258453173}{344160520704000}}\,{n}^{4}-{ \frac {21314536118427329}{235382809559040}}\,{n}^{11}-{\frac { 9413144941325}{963555360768}}\,{n}^{12}-{\frac {100028650865917027}{ 106198103531520}}\,{n}^{8}-{\frac {50082578366831567339}{ 67966786260172800}}\,{n}^{9}+{\frac {171783}{20}}\,n+{\frac { 1966529001671307859267}{198236459925504000}}\,{n}^{7}+{\frac { 403598464050770774633}{4129926248448000}}\,{n}^{5}+{\frac { 2364909207410262013187}{49559114981376000}}\,{n}^{6} Theorem Number, 10, : the , 10, -th factorial moment of X is / 101645143274263 25 167209283471 26 |-362880 - --------------------- a - -------------------- a \ 262623662109307699200 26262366210930769920 21026753291 27 123592657 25 - --------------------- a - ----------------- a n 472722591796753858560 54037790557470720 16531112606381 24 13399903 23 2 - -------------------- a n - --------------- a n 52524732421861539840 178637324156928 29914537600951 23 716925665 22 2 905213 21 3 - ------------------- a n - -------------- a n - ------------ a n 1641397888183173120 78315638489088 579877208064 9769947127205 21 2 2086838399 20 3 3839961 19 4 - ----------------- a n - -------------- a n - ------------ a n 20760763803107328 12205034569728 159542935552 27678419 19 3 259032595 18 4 21913 17 5 - ---------- a n - ------------ a n - -------- a n 3556245504 113359454208 80740352 12796923835 17 4 50814637 16 5 159017 15 6 - ------------ a n - ---------- a n - -------- a n 142748942336 2290089984 69206016 96937369 15 5 4116755 14 6 3566645 13 7 - --------- a n - -------- a n - --------- a n 131203072 25952256 242221056 11825332195 13 6 401355205 12 7 131095925 11 8 - ----------- a n - --------- a n - ---------- a n 2699034624 484442112 1937768448 1119667435 11 7 273707935 10 8 30267175 9 9 - ---------- a n - --------- a n - --------- a n 60555264 88080384 132120576 9552584215 9 8 474062075 8 9 16145585 10 7 1732397455 9 7 - ---------- a n - --------- a n - -------- n a - ---------- n a 176160768 58720256 29360128 16515072 41870365 10 6 16145585 11 5 4766577091 10 5 629677815 11 4 - -------- n a - -------- n a - 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