For #12 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k - 1) G[n](q) = ) q G[k - 1](q) G[n - k](q) / ----- k = 1 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 1/2 3 2 1/2 1/2 Pi m 3 m 9 Pi m 17 Pi -------- - ---- + --------- - 1/2 + -------- 2 2 16 256 m and in Maple input form 1/2*Pi^(1/2)*m^3-3/2*m^2+9/16*Pi^(1/2)*m-1/2+17/256*Pi^(1/2)/m and in floating-point: .8862269255*m^3-1.500000000*m^2+.9970052912*m-.5000000000+.1177020135/m The asymoptotics of the Variance to order, 4, is: / Pi \ 6 / 9 Pi \ 4 / 49 Pi \ 2 |- ---- + 5/6| m + |- ---- + 7/4| m + |- ----- + 7/6| m \ 4 / \ 16 / \ 128 / and in Maple input form (-1/4*Pi+5/6)*m^6+(-9/16*Pi+7/4)*m^4+(-49/128*Pi+7/6)*m^2 and in floating-point: .479351698e-1*m^6-.17145868e-1*m^4-.35974271e-1*m^2 The asympotic skewness squared is, 0.4907980946 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 2 27 Pi (64 Pi - 400 Pi + 625) 81 Pi (216 Pi - 1355 Pi + 2125) - ----------------------------- + -------------------------------- 3 4 2 16 (3 Pi - 10) 32 (3 Pi - 10) m 3 2 243 Pi (2616 Pi - 23342 Pi + 69205 Pi - 68150) - ------------------------------------------------ 5 4 256 (3 Pi - 10) m and in Maple input form -27/16*Pi*(64*Pi^2-400*Pi+625)/(3*Pi-10)^3+81/32*Pi*(216*Pi^2-1355*Pi+2125)/(3* Pi-10)^4/m^2-243/256*Pi*(2616*Pi^3-23342*Pi^2+69205*Pi-68150)/(3*Pi-10)^5/m^4 and in floating-point 1.706547169 1.582504831 0.4907980946 - ----------- + ----------- 2 4 m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 189 Pi - 315 Pi - 884 - ---------------------- 2 7 (3 Pi - 10) and in floating-point 3.560394751 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 2 189 Pi - 315 Pi - 884 3 (8505 Pi - 37962 Pi + 35360) - ---------------------- + ------------------------------- 2 3 2 7 (3 Pi - 10) 140 (3 Pi - 10) m 3 2 3 (309015 Pi - 2330424 Pi + 5681532 Pi - 4430080) - --------------------------------------------------- 4 4 1120 (3 Pi - 10) m and in Maple input form -1/7*(189*Pi^2-315*Pi-884)/(3*Pi-10)^2+3/140*(8505*Pi^2-37962*Pi+35360)/(3*Pi-\ 10)^3/m^2-3/1120*(309015*Pi^3-2330424*Pi^2+5681532*Pi-4430080)/(3*Pi-10)^4/m^4 and in floating-point 4.486025107 0.5092576731 3.560394751 - ----------- - ------------ 2 4 m m The square of the limit of the, 5, -th standarized moment is: - 3 Pi ( 4 3 2 65028096 Pi + 135475200 Pi - 1636508160 Pi - 1778196000 Pi + 11203164025 / 5 ) / (50176 (3 Pi - 10) ) / and in floating-point 348.9092799 758.8817150 52.64319243 - ----------- + ----------- 2 4 m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: - 3 Pi ( 4 3 2 65028096 Pi + 135475200 Pi - 1636508160 Pi - 1778196000 Pi + 11203164025 / 5 4 3 ) / (50176 (3 Pi - 10) ) + 9 Pi (1097349120 Pi - 3476100096 Pi / 2 / 6 - 15536591280 Pi + 64461199830 Pi - 48281725975) / (100352 (3 Pi - 10) / 2 5 4 3 m ) - 9 Pi (39870351360 Pi - 178698949632 Pi - 694762956720 Pi 2 / + 5236914521448 Pi - 9588924190395 Pi + 5186021991950) / (802816 / 7 4 (3 Pi - 10) m ) and in Maple input form -3/50176*Pi*(65028096*Pi^4+135475200*Pi^3-1636508160*Pi^2-1778196000*Pi+ 11203164025)/(3*Pi-10)^5+9/100352*Pi*(1097349120*Pi^4-3476100096*Pi^3-\ 15536591280*Pi^2+64461199830*Pi-48281725975)/(3*Pi-10)^6/m^2-9/802816*Pi*( 39870351360*Pi^5-178698949632*Pi^4-694762956720*Pi^3+5236914521448*Pi^2-\ 9588924190395*Pi+5186021991950)/(3*Pi-10)^7/m^4 and in floating-point 348.9092799 758.8817150 52.64319243 - ----------- + ----------- 2 4 m m The limit of the, 6, -th standarized moment is: , 3 2 15 (144144 Pi + 720720 Pi - 3013725 Pi - 2120320) --------------------------------------------------- 3 16016 (3 Pi - 10) and in floating-point 27.68549546 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 15 (144144 Pi + 720720 Pi - 3013725 Pi - 2120320) --------------------------------------------------- 3 16016 (3 Pi - 10) 3 2 27 (3243240 Pi - 7737873 Pi - 21194115 Pi + 42406400) - ------------------------------------------------------- + 9 ( 4 2 32032 (3 Pi - 10) m 4 3 2 353513160 Pi - 1485859518 Pi - 2914157367 Pi + 19796615250 Pi / 5 4 - 21795942400) / (256256 (3 Pi - 10) m ) / and in Maple input form 15/16016*(144144*Pi^3+720720*Pi^2-3013725*Pi-2120320)/(3*Pi-10)^3-27/32032*( 3243240*Pi^3-7737873*Pi^2-21194115*Pi+42406400)/(3*Pi-10)^4/m^2+9/256256*( 353513160*Pi^4-1485859518*Pi^3-2914157367*Pi^2+19796615250*Pi-21795942400)/(3* Pi-10)^5/m^4 and in floating-point 109.1371626 110.1209671 27.68549546 - ----------- + ----------- 2 4 m m This ends this exciting article that took, 6.354, seconds to produce ------------------------------------------ For #21 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k (n - k)) G[n](q) = ) q G[k - 1](q) G[n - k](q) / ----- k = 1 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 4 1/2 3 1/2 m Pi m 2 9 Pi m ---- - -------- + m - --------- + 1/2 2 2 16 and in Maple input form 1/2*m^4-1/2*Pi^(1/2)*m^3+m^2-9/16*Pi^(1/2)*m+1/2 and in floating-point: .5000000000*m^4-.8862269255*m^3+m^2-.9970052912*m+.5000000000 The asymoptotics of the Variance to order, 4, is: / Pi \ 6 / 9 Pi \ 4 / 49 Pi \ 2 |- ---- + 5/6| m + |- ---- + 7/4| m + |- ----- + 7/6| m \ 4 / \ 16 / \ 128 / and in Maple input form (-1/4*Pi+5/6)*m^6+(-9/16*Pi+7/4)*m^4+(-49/128*Pi+7/6)*m^2 and in floating-point: .479351698e-1*m^6-.17145868e-1*m^4-.35974271e-1*m^2 The asympotic skewness squared is, 0.4907980946 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 2 27 Pi (64 Pi - 400 Pi + 625) 81 Pi (216 Pi - 1355 Pi + 2125) - ----------------------------- + -------------------------------- 3 4 2 16 (3 Pi - 10) 32 (3 Pi - 10) m 3 2 243 Pi (2616 Pi - 23342 Pi + 69205 Pi - 68150) - ------------------------------------------------ 5 4 256 (3 Pi - 10) m and in Maple input form -27/16*Pi*(64*Pi^2-400*Pi+625)/(3*Pi-10)^3+81/32*Pi*(216*Pi^2-1355*Pi+2125)/(3* Pi-10)^4/m^2-243/256*Pi*(2616*Pi^3-23342*Pi^2+69205*Pi-68150)/(3*Pi-10)^5/m^4 and in floating-point 1.706547169 1.582504831 0.4907980946 - ----------- + ----------- 2 4 m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 189 Pi - 315 Pi - 884 - ---------------------- 2 7 (3 Pi - 10) and in floating-point 3.560394751 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 2 189 Pi - 315 Pi - 884 3 (8505 Pi - 37962 Pi + 35360) - ---------------------- + ------------------------------- 2 3 2 7 (3 Pi - 10) 140 (3 Pi - 10) m 3 2 3 (309015 Pi - 2330424 Pi + 5681532 Pi - 4430080) - --------------------------------------------------- 4 4 1120 (3 Pi - 10) m and in Maple input form -1/7*(189*Pi^2-315*Pi-884)/(3*Pi-10)^2+3/140*(8505*Pi^2-37962*Pi+35360)/(3*Pi-\ 10)^3/m^2-3/1120*(309015*Pi^3-2330424*Pi^2+5681532*Pi-4430080)/(3*Pi-10)^4/m^4 and in floating-point 4.486025107 0.5092576731 3.560394751 - ----------- - ------------ 2 4 m m The square of the limit of the, 5, -th standarized moment is: - 3 Pi ( 4 3 2 65028096 Pi + 135475200 Pi - 1636508160 Pi - 1778196000 Pi + 11203164025 / 5 ) / (50176 (3 Pi - 10) ) / and in floating-point 348.9092799 758.8817150 52.64319243 - ----------- + ----------- 2 4 m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: - 3 Pi ( 4 3 2 65028096 Pi + 135475200 Pi - 1636508160 Pi - 1778196000 Pi + 11203164025 / 5 4 3 ) / (50176 (3 Pi - 10) ) + 9 Pi (1097349120 Pi - 3476100096 Pi / 2 / 6 - 15536591280 Pi + 64461199830 Pi - 48281725975) / (100352 (3 Pi - 10) / 2 5 4 3 m ) - 9 Pi (39870351360 Pi - 178698949632 Pi - 694762956720 Pi 2 / + 5236914521448 Pi - 9588924190395 Pi + 5186021991950) / (802816 / 7 4 (3 Pi - 10) m ) and in Maple input form -3/50176*Pi*(65028096*Pi^4+135475200*Pi^3-1636508160*Pi^2-1778196000*Pi+ 11203164025)/(3*Pi-10)^5+9/100352*Pi*(1097349120*Pi^4-3476100096*Pi^3-\ 15536591280*Pi^2+64461199830*Pi-48281725975)/(3*Pi-10)^6/m^2-9/802816*Pi*( 39870351360*Pi^5-178698949632*Pi^4-694762956720*Pi^3+5236914521448*Pi^2-\ 9588924190395*Pi+5186021991950)/(3*Pi-10)^7/m^4 and in floating-point 348.9092799 758.8817150 52.64319243 - ----------- + ----------- 2 4 m m The limit of the, 6, -th standarized moment is: , 3 2 15 (144144 Pi + 720720 Pi - 3013725 Pi - 2120320) --------------------------------------------------- 3 16016 (3 Pi - 10) and in floating-point 27.68549546 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 15 (144144 Pi + 720720 Pi - 3013725 Pi - 2120320) --------------------------------------------------- 3 16016 (3 Pi - 10) 3 2 27 (3243240 Pi - 7737873 Pi - 21194115 Pi + 42406400) - ------------------------------------------------------- + 9 ( 4 2 32032 (3 Pi - 10) m 4 3 2 353513160 Pi - 1485859518 Pi - 2914157367 Pi + 19796615250 Pi / 5 4 - 21795942400) / (256256 (3 Pi - 10) m ) / and in Maple input form 15/16016*(144144*Pi^3+720720*Pi^2-3013725*Pi-2120320)/(3*Pi-10)^3-27/32032*( 3243240*Pi^3-7737873*Pi^2-21194115*Pi+42406400)/(3*Pi-10)^4/m^2+9/256256*( 353513160*Pi^4-1485859518*Pi^3-2914157367*Pi^2+19796615250*Pi-21795942400)/(3* Pi-10)^5/m^4 and in floating-point 109.1371626 110.1209671 27.68549546 - ----------- + ----------- 2 4 m m This ends this exciting article that took, 15.862, seconds to produce ------------------------------------------ For #123 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](t, q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k - 1) G[n](t, q) = ) q G[k - 1](t, q t) G[n - k](t, q) / ----- k = 1 Note that what we REALLY care about is the statistic carried by the variable, t, and the remaining variable , q serves as a CATALYTIC variable, to enable the recurrence. At the end of the day we are really only interested in the random variable defined on the n-th set of our family whose weight-enumerator is: G(t, 1)(n) 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 4 2 1/2 m 1/2 3 5 m 9 Pi m ---- - Pi m + ---- - --------- + 1 2 2 8 and in Maple input form 1/2*m^4-Pi^(1/2)*m^3+5/2*m^2-9/8*Pi^(1/2)*m+1 and in floating-point: .5000000000*m^4-1.772453851*m^3+2.500000000*m^2-1.994010582*m+1. The asymoptotics of the Variance to order, 4, is: 8 1/2 7 1/2 5 m Pi m / 101\ 6 Pi m / 9 Pi 104\ 4 ---- - -------- + |-Pi + ---| m - -------- + |- ---- + ---| m 15 8 \ 30 / 64 \ 4 15 / and in Maple input form 1/15*m^8-1/8*Pi^(1/2)*m^7+(-Pi+101/30)*m^6-1/64*Pi^(1/2)*m^5+(-9/4*Pi+104/15)*m ^4 and in floating-point: .6666666667e-1*m^8-.2215567314*m^7+.225074013*m^6-.2769459142e-1*m^5-.135250139 *m^4 The asympotic skewness squared is, 2.176870748 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 3/2 1/2 28198335 Pi 5360 1743862125 Pi 191085 Pi 1/2 ----------- - ---- - ---------------- + ------------ 320 915 Pi 802816 49 6422528 224 --- + --------- + ------------------ + --------------------------------- 147 392 m 2 3 m m 14530399875 2 105355 576447045 - ----------- Pi + ------ + --------- Pi 25690112 21 3211264 + ----------------------------------------- 4 m and in Maple input form 320/147+915/392*Pi^(1/2)/m+(28198335/802816*Pi-5360/49)/m^2+(-1743862125/ 6422528*Pi^(3/2)+191085/224*Pi^(1/2))/m^3+(-14530399875/25690112*Pi^2+105355/21 +576447045/3211264*Pi)/m^4 and in floating-point 4.137232840 0.9584289 0.078565 1.4308604 2.176870748 + ----------- + --------- + -------- - --------- m 2 3 4 m m m The limit of the kurtosis (aka 4th standarized moment) is: , 45/7 and in floating-point 6.428571429 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 3/2 1/2 1214205 Pi 10470 124693575 Pi 76635775 Pi 1/2 ---------- - ----- - --------------- + -------------- 26995 Pi 28672 77 458752 90112 45/7 + ----------- + ------------------ + ---------------------------------- 7168 m 2 3 m m 2046946275 2 8939450085 5605 - ---------- Pi + ---------- Pi - ---- 917504 1261568 22 + --------------------------------------- 4 m and in Maple input form 45/7+26995/7168*Pi^(1/2)/m+(1214205/28672*Pi-10470/77)/m^2+(-124693575/458752* Pi^(3/2)+76635775/90112*Pi^(1/2))/m^3+(-2046946275/917504*Pi^2+8939450085/ 1261568*Pi-5605/22)/m^4 and in floating-point 6.675138352 2.9335158 6.145725 12.5309273 6.428571429 + ----------- - --------- - -------- - ---------- m 2 3 4 m m m The square of the limit of the, 5, -th standarized moment is: 11552000 -------- 17787 and in floating-point 2363.714248 1117.44179 6439.1339 15073.969 649.4630910 + ----------- + ---------- - --------- - --------- m 2 3 4 m m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: 100116049759059375 Pi 48336000 1/2 --------------------- - -------- 11552000 8096576875 Pi 6366215274496 1001 -------- + ---------------- + -------------------------------- 17787 6071296 m 2 m 3/2 1/2 4688084231535826875 Pi 25757295640625 Pi - ------------------------- + -------------------- 50929722195968 90202112 + -------------------------------------------------- 3 m 3796748728000 301014984053064009375 2 633671971693788973125 ------------- - --------------------- Pi + --------------------- Pi 3006003 407437777567744 331043194273792 + -------------------------------------------------------------------- 4 m and in Maple input form 11552000/17787+8096576875/6071296*Pi^(1/2)/m+(100116049759059375/6366215274496* Pi-48336000/1001)/m^2+(-4688084231535826875/50929722195968*Pi^(3/2)+ 25757295640625/90202112*Pi^(1/2))/m^3+(3796748728000/3006003-\ 301014984053064009375/407437777567744*Pi^2+633671971693788973125/ 331043194273792*Pi)/m^4 and in floating-point 2363.714248 1117.44179 6439.1339 15073.969 649.4630910 + ----------- + ---------- - --------- - --------- m 2 3 4 m m m 912825 The limit of the, 6, -th standarized moment is: , ------ 7007 and in floating-point 130.2732981 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 27571317310125 Pi 13626498000 1/2 ----------------- - ----------- 912825 29755891875 Pi 14694744064 2263261 ------ + ----------------- + ------------------------------- 7007 166985728 m 2 m 3/2 1/2 731024275696125 Pi 16601546481833325 Pi - --------------------- + ----------------------- 58778976256 431491121152 + ------------------------------------------------- 3 m 54340752646351125 2 983404876719983625 623909025 - ----------------- Pi + ------------------ Pi - --------- 470231810048 2712229904384 4526522 + ----------------------------------------------------------- 4 m and in Maple input form 912825/7007+29755891875/166985728*Pi^(1/2)/m+(27571317310125/14694744064*Pi-\ 13626498000/2263261)/m^2+(-731024275696125/58778976256*Pi^(3/2)+ 16601546481833325/431491121152*Pi^(1/2))/m^3+(-54340752646351125/470231810048* Pi^2+983404876719983625/2712229904384*Pi-623909025/4526522)/m^4 and in floating-point 315.8410348 126.258294 1057.50572 1601.425086 130.2732981 + ----------- - ---------- - ---------- - ----------- m 2 3 4 m m m This ends this exciting article that took, 48.069, seconds to produce ------------------------------------------ For #321 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](t, q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k (n - k)) (n - k) k G[n](t, q) = ) q G[k - 1](t, q t ) G[n - k](t, t q) / ----- k = 1 Note that what we REALLY care about is the statistic carried by the variable, t, and the remaining variable , q serves as a CATALYTIC variable, to enable the recurrence. At the end of the day we are really only interested in the random variable defined on the n-th set of our family whose weight-enumerator is: G(t, 1)(n) 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 6 1/2 5 4 1/2 3 2 m 3 Pi m 5 m 59 Pi m 19 m ---- - ---------- + ---- - ----------- + ----- 6 8 4 64 12 and in Maple input form 1/6*m^6-3/8*Pi^(1/2)*m^5+5/4*m^4-59/64*Pi^(1/2)*m^3+19/12*m^2 and in floating-point: .1666666667*m^6-.6646701941*m^5+1.250000000*m^4-1.633980894*m^3+1.583333333*m^2 The asymoptotics of the Variance to order, 4, is: 1/2 9 1/2 7 / 9 Pi 11\ 10 7 Pi m / 177 Pi 547\ 8 29 Pi m |- ---- + --| m - ---------- + |- ------ + ---| m - ----------- \ 64 24/ 192 \ 256 240/ 512 / 2681 Pi 83\ 6 + |- ------- + --| m \ 2048 20/ and in Maple input form (-9/64*Pi+11/24)*m^10-7/192*Pi^(1/2)*m^9+(-177/256*Pi+547/240)*m^8-29/512*Pi^(1 /2)*m^7+(-2681/2048*Pi+83/20)*m^6 and in floating-point: .165468663e-1*m^10-.6462071331e-1*m^9+.107049871*m^8-.1003928939*m^7+.37397508e\ -1*m^6 The asympotic skewness squared is, 0.4061523062 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 3 Pi (167961600 Pi - 1052896320 Pi + 1650065641) - ------------------------------------------------- 3 6400 (27 Pi - 88) 1/2 2 3 Pi (137424587040 Pi - 862839950529 Pi + 1354362634240) + ------------------------------------------------------------ - 3 ( 4 70400 (27 Pi - 88) m 4 3 2 108113254999200 Pi - 900006809693805 Pi + 2735565926062056 Pi / 5 2 - 3948137149153824 Pi + 2779129053184000) / (7744000 (27 Pi - 88) m ) / 1/2 4 3 - Pi (3105580642037517600 Pi - 40661640516600239895 Pi 2 + 198767281844130281088 Pi - 430118362343095284672 Pi / 6 3 + 347756534839173447680) / (216832000 (27 Pi - 88) m ) + ( / 6 5 315564336349156818000 Pi - 4812227384449572704925 Pi 4 3 + 29812067866627558255350 Pi - 95921791564242222261288 Pi 2 + 169581042060421825705344 Pi - 158463450739563649287168 Pi / 7 4 + 63597154206698438656000) / (1084160000 (27 Pi - 88) m ) / and in Maple input form -3/6400*Pi*(167961600*Pi^2-1052896320*Pi+1650065641)/(27*Pi-88)^3+3/70400*Pi^(1 /2)*(137424587040*Pi^2-862839950529*Pi+1354362634240)/(27*Pi-88)^4/m-3/7744000* (108113254999200*Pi^4-900006809693805*Pi^3+2735565926062056*Pi^2-\ 3948137149153824*Pi+2779129053184000)/(27*Pi-88)^5/m^2-1/216832000*Pi^(1/2)*( 3105580642037517600*Pi^4-40661640516600239895*Pi^3+198767281844130281088*Pi^2-\ 430118362343095284672*Pi+347756534839173447680)/(27*Pi-88)^6/m^3+1/1084160000*( 315564336349156818000*Pi^6-4812227384449572704925*Pi^5+29812067866627558255350* Pi^4-95921791564242222261288*Pi^3+169581042060421825705344*Pi^2-\ 158463450739563649287168*Pi+63597154206698438656000)/(27*Pi-88)^7/m^4 and in floating-point 2.006983657 2.825966166 17.48929544 279.5264438 0.4061523062 - ----------- - ----------- + ----------- - ----------- m 2 3 4 m m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 3 (6633900 Pi - 12204465 Pi - 27241216) - ---------------------------------------- 2 9100 (27 Pi - 88) and in floating-point 3.550545867 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 3 (6633900 Pi - 12204465 Pi - 27241216) - ---------------------------------------- 2 9100 (27 Pi - 88) 1/2 3 Pi (17994219555 Pi - 56521598936) + -------------------------------------- - ( 3 400400 (27 Pi - 88) m 3 2 3105525948525 Pi - 23864817334230 Pi + 66581571543696 Pi - 69925303681024 / 4 2 1/2 3 ) / (2002000 (27 Pi - 88) m ) - Pi (5101438391439975 Pi / 2 / - 50147044913341440 Pi + 163515119280851712 Pi - 176942951455221248) / / 5 3 5 (16016000 (27 Pi - 88) m ) + (863286273682457625 Pi 4 3 - 12453922899637263000 Pi + 72555918878431622880 Pi 2 - 214575507451845779712 Pi + 324011492333732222976 Pi / 6 4 - 200882928037692178432) / (80080000 (27 Pi - 88) m ) / and in Maple input form -3/9100*(6633900*Pi^2-12204465*Pi-27241216)/(27*Pi-88)^2+3/400400*Pi^(1/2)*( 17994219555*Pi-56521598936)/(27*Pi-88)^3/m-1/2002000*(3105525948525*Pi^3-\ 23864817334230*Pi^2+66581571543696*Pi-69925303681024)/(27*Pi-88)^4/m^2-1/ 16016000*Pi^(1/2)*(5101438391439975*Pi^3-50147044913341440*Pi^2+ 163515119280851712*Pi-176942951455221248)/(27*Pi-88)^5/m^3+1/80080000*( 863286273682457625*Pi^5-12453922899637263000*Pi^4+72555918878431622880*Pi^3-\ 214575507451845779712*Pi^2+324011492333732222976*Pi-200882928037692178432)/(27* Pi-88)^6/m^4 and in floating-point 3.689623633 6.682882587 20.89186193 21.85981658 3.550545867 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The square of the limit of the, 5, -th standarized moment is: 4 3 - 3 Pi (84758679886233600 Pi + 105882949641830400 Pi 2 - 1971867633886126080 Pi - 1252311260069151360 Pi + 11856505105541519641) / 5 / (1107558400 (27 Pi - 88) ) / and in floating-point 7 312.8838384 67.54576207 16486.71429 0.3781051371 10 46.53640377 - ----------- - ----------- - ----------- - ---------------- m 2 3 4 m m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: 4 3 - 3 Pi (84758679886233600 Pi + 105882949641830400 Pi 2 - 1971867633886126080 Pi - 1252311260069151360 Pi + 11856505105541519641) / 5 1/2 4 / (1107558400 (27 Pi - 88) ) + Pi (12011933414870576706355200 Pi / 3 2 + 37834909386288578280437760 Pi - 336990287929603912082856000 Pi / - 492331684293901444081058829 Pi + 2529475141689372421501485056) / ( / 6 6 578467784294400 (27 Pi - 88) m) - (553484358009792114236348335718400 Pi 5 - 4009628328893685224131293546577920 Pi 4 + 4082644109957311635968412182383200 Pi 3 + 46956536028259848546231480126300153 Pi 2 - 174523654868973478703425610699566671 Pi + 136506159295945131613871588985903600 Pi / + 134910001629275329043828228881383424) / (906385552580719411200 / 7 2 1/2 6 (27 Pi - 88) m ) - Pi (43460835855869311228096636118630400 Pi 5 - 143238415646991039070424902954260480 Pi 4 - 1893828139378074530420510286721995840 Pi 3 + 9004240761838883921143555404347116125 Pi 2 + 6882776482025615724629378010823480056 Pi - 85004056978596595794076558287897203136 Pi / + 106456933993498177879037743336572059648) / (1035869202949393612800 / 8 3 8 (27 Pi - 88) m ) + (38464937734197961288784901354743808000 Pi 7 - 535278226314204567470274272860339353600 Pi 6 + 2532054578806287018627723460581785065800 Pi 5 - 1577000038064781306352035383689201702515 Pi 4 - 29040653503574230225620427958004522025995 Pi 3 + 114467101979614355546998181914681260626076 Pi 2 - 165775183392723381159980110612781494219488 Pi + 45922534104212324938647046993183317655296 Pi / + 71515202011977879090491977113837544407040) / (9063855525807194112000 / 9 4 (27 Pi - 88) m ) and in Maple input form -3/1107558400*Pi*(84758679886233600*Pi^4+105882949641830400*Pi^3-\ 1971867633886126080*Pi^2-1252311260069151360*Pi+11856505105541519641)/(27*Pi-88 )^5+1/578467784294400*Pi^(1/2)*(12011933414870576706355200*Pi^4+ 37834909386288578280437760*Pi^3-336990287929603912082856000*Pi^2-\ 492331684293901444081058829*Pi+2529475141689372421501485056)/(27*Pi-88)^6/m-1/ 906385552580719411200*(553484358009792114236348335718400*Pi^6-\ 4009628328893685224131293546577920*Pi^5+4082644109957311635968412182383200*Pi^4 +46956536028259848546231480126300153*Pi^3-174523654868973478703425610699566671* Pi^2+136506159295945131613871588985903600*Pi+ 134910001629275329043828228881383424)/(27*Pi-88)^7/m^2-1/1035869202949393612800 *Pi^(1/2)*(43460835855869311228096636118630400*Pi^6-\ 143238415646991039070424902954260480*Pi^5-1893828139378074530420510286721995840 *Pi^4+9004240761838883921143555404347116125*Pi^3+ 6882776482025615724629378010823480056*Pi^2-\ 85004056978596595794076558287897203136*Pi+ 106456933993498177879037743336572059648)/(27*Pi-88)^8/m^3+1/ 9063855525807194112000*(38464937734197961288784901354743808000*Pi^8-\ 535278226314204567470274272860339353600*Pi^7+ 2532054578806287018627723460581785065800*Pi^6-\ 1577000038064781306352035383689201702515*Pi^5-\ 29040653503574230225620427958004522025995*Pi^4+ 114467101979614355546998181914681260626076*Pi^3-\ 165775183392723381159980110612781494219488*Pi^2+ 45922534104212324938647046993183317655296*Pi+ 71515202011977879090491977113837544407040)/(27*Pi-88)^9/m^4 and in floating-point 7 312.8838384 67.54576207 16486.71429 0.3781051371 10 46.53640377 - ----------- - ----------- - ----------- - ---------------- m 2 3 4 m m m 3 The limit of the, 6, -th standarized moment is: , (491807339543520000 Pi 2 + 2094431379944904000 Pi - 9744590555461914075 Pi - 5311173605528305664) / 3 / (4997280288000 (27 Pi - 88) ) / and in floating-point 27.35868907 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 (491807339543520000 Pi + 2094431379944904000 Pi - 9744590555461914075 Pi / 3 1/2 - 5311173605528305664) / (4997280288000 (27 Pi - 88) ) + Pi ( / 2 13188737969234100000 Pi - 346613084360011862775 Pi + 958732682799788953096 / 4 4 ) / (3331520192000 (27 Pi - 88) m) + (3488327983464744420000 Pi / 3 2 - 20034515495169467200125 Pi + 3500174299397329206225 Pi / + 195577088448910746329432 Pi - 367567456336486688882688) / ( / 5 2 1/2 4 16657600960000 (27 Pi - 88) m ) - Pi (5686521656898093675300000 Pi 3 2 - 141980144603450023700236125 Pi + 1032869489806883614840137000 Pi / - 2977095289720119232429089856 Pi + 3007163982398001755364862464) / ( / 6 3 6 399782423040000 (27 Pi - 88) m ) - (1454548625326006541481150000 Pi 5 4 - 17676236480608297868229095625 Pi + 67418032684300209022703511375 Pi 3 2 + 8777976208119221858239869720 Pi - 725614939246377534350741044608 Pi + 1875222637758825684817045983744 Pi - 1558065919754736508175151792128) / 7 4 / (999456057600000 (27 Pi - 88) m ) / and in Maple input form 1/4997280288000*(491807339543520000*Pi^3+2094431379944904000*Pi^2-\ 9744590555461914075*Pi-5311173605528305664)/(27*Pi-88)^3+1/3331520192000*Pi^(1/ 2)*(13188737969234100000*Pi^2-346613084360011862775*Pi+958732682799788953096)/( 27*Pi-88)^4/m+1/16657600960000*(3488327983464744420000*Pi^4-\ 20034515495169467200125*Pi^3+3500174299397329206225*Pi^2+ 195577088448910746329432*Pi-367567456336486688882688)/(27*Pi-88)^5/m^2-1/ 399782423040000*Pi^(1/2)*(5686521656898093675300000*Pi^4-\ 141980144603450023700236125*Pi^3+1032869489806883614840137000*Pi^2-\ 2977095289720119232429089856*Pi+3007163982398001755364862464)/(27*Pi-88)^6/m^3-\ 1/999456057600000*(1454548625326006541481150000*Pi^6-\ 17676236480608297868229095625*Pi^5+67418032684300209022703511375*Pi^4+ 8777976208119221858239869720*Pi^3-725614939246377534350741044608*Pi^2+ 1875222637758825684817045983744*Pi-1558065919754736508175151792128)/(27*Pi-88)^ 7/m^4 and in floating-point 87.79106805 102.1636823 801.9776356 306.2791125 27.35868907 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m This ends this exciting article that took, 1646.724, seconds to produce ------------------------------------------ For #213 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](t, q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k (n - k)) G[n](t, q) = ) q G[k - 1](t, q t) G[n - k](t, q) / ----- k = 1 Note that what we REALLY care about is the statistic carried by the variable, t, and the remaining variable , q serves as a CATALYTIC variable, to enable the recurrence. At the end of the day we are really only interested in the random variable defined on the n-th set of our family whose weight-enumerator is: G(t, 1)(n) 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 1/2 5 4 1/2 3 2 1/2 Pi m 3 m 41 Pi m 5 m 593 Pi m -------- - ---- + ----------- - ---- + ----------- 8 4 64 4 1024 and in Maple input form 1/8*Pi^(1/2)*m^5-3/4*m^4+41/64*Pi^(1/2)*m^3-5/4*m^2+593/1024*Pi^(1/2)*m and in floating-point: .2215567314*m^5-.7500000000*m^4+1.135478248*m^3-1.250000000*m^2+1.026430795*m The asymoptotics of the Variance to order, 4, is: 1/2 9 1/2 7 / Pi \ 10 5 Pi m / 41 Pi 49\ 8 39 Pi m |- ---- + 7/120| m - ---------- + |- ----- + --| m - ----------- \ 64 / 192 \ 256 80/ 512 / 1137 Pi 109\ 6 + |- ------- + ---| m \ 2048 60 / and in Maple input form (-1/64*Pi+7/120)*m^10-5/192*Pi^(1/2)*m^9+(-41/256*Pi+49/80)*m^8-39/512*Pi^(1/2) *m^7+(-1137/2048*Pi+109/60)*m^6 and in floating-point: .924594811e-2*m^10-.4615765238e-1*m^9+.1093543015*m^8-.1350111332*m^7+.72530511\ e-1*m^6 The asympotic skewness squared is, 0.5321966823 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 15 Pi (230400 Pi - 1404480 Pi + 2140369) - ----------------------------------------- 3 256 (15 Pi - 56) 1/2 2 125 Pi (714008160 Pi - 4433297211 Pi + 6879330304) + ------------------------------------------------------ - 25 ( 4 19712 (15 Pi - 56) m 4 3 2 5706510717600 Pi - 6213161094525 Pi + 12438717027624 Pi / 5 2 - 594587975426592 Pi + 1381922504704000) / (9106944 (15 Pi - 56) m ) - / 1/2 4 3 15 Pi (5703806285100000 Pi - 85936207872986625 Pi 2 + 446964294155862400 Pi - 971470042702859840 Pi + 749557237375041536) / 6 3 6 / (12142592 (15 Pi - 56) m ) + 5 (613973685713850000 Pi / 5 4 - 7514460969376370625 Pi + 37198401055917018750 Pi 3 2 - 146018810511429165000 Pi + 585021893245515358080 Pi / - 1433385241058788512768 Pi + 1342519620246752460800) / (36427776 / 7 4 (15 Pi - 56) m ) and in Maple input form -15/256*Pi*(230400*Pi^2-1404480*Pi+2140369)/(15*Pi-56)^3+125/19712*Pi^(1/2)*( 714008160*Pi^2-4433297211*Pi+6879330304)/(15*Pi-56)^4/m-25/9106944*( 5706510717600*Pi^4-6213161094525*Pi^3+12438717027624*Pi^2-594587975426592*Pi+ 1381922504704000)/(15*Pi-56)^5/m^2-15/12142592*Pi^(1/2)*(5703806285100000*Pi^4-\ 85936207872986625*Pi^3+446964294155862400*Pi^2-971470042702859840*Pi+ 749557237375041536)/(15*Pi-56)^6/m^3+5/36427776*(613973685713850000*Pi^6-\ 7514460969376370625*Pi^5+37198401055917018750*Pi^4-146018810511429165000*Pi^3+ 585021893245515358080*Pi^2-1433385241058788512768*Pi+1342519620246752460800)/( 15*Pi-56)^7/m^4 and in floating-point 2.364084609 2.315104037 19.69530778 24.88589613 0.5321966823 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 1158300 Pi - 765765 Pi - 9492224 - --------------------------------- 2 1716 (15 Pi - 56) and in floating-point 3.446730626 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 1/2 1158300 Pi - 765765 Pi - 9492224 125 Pi (70581069 Pi - 221020520) - --------------------------------- + ----------------------------------- 2 3 1716 (15 Pi - 56) 48048 (15 Pi - 56) m 3 2 5 (10035800775 Pi - 64625678730 Pi + 338031793008 Pi - 734766104576) - ---------------------------------------------------------------------- 4 2 48048 (15 Pi - 56) m 1/2 3 2 - 25 Pi (4189974162075 Pi - 46628713244160 Pi + 158247148525312 Pi / 5 3 5 - 166841377644032) / (384384 (15 Pi - 56) m ) + (2159539478971875 Pi / 4 3 2 - 32555911818465000 Pi + 219366946123812000 Pi - 858571100109049600 Pi / 6 + 1866698297036482560 Pi - 1682010685019521024) / (384384 (15 Pi - 56) / 4 m ) and in Maple input form -1/1716*(1158300*Pi^2-765765*Pi-9492224)/(15*Pi-56)^2+125/48048*Pi^(1/2)*( 70581069*Pi-221020520)/(15*Pi-56)^3/m-5/48048*(10035800775*Pi^3-64625678730*Pi^ 2+338031793008*Pi-734766104576)/(15*Pi-56)^4/m^2-25/384384*Pi^(1/2)*( 4189974162075*Pi^3-46628713244160*Pi^2+158247148525312*Pi-166841377644032)/(15* Pi-56)^5/m^3+1/384384*(2159539478971875*Pi^5-32555911818465000*Pi^4+ 219366946123812000*Pi^3-858571100109049600*Pi^2+1866698297036482560*Pi-\ 1682010685019521024)/(15*Pi-56)^6/m^4 and in floating-point 4.724177355 8.969710599 31.90094348 52.66641812 3.446730626 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The square of the limit of the, 5, -th standarized moment is: 4 3 2 - 125 Pi (1563145902489600 Pi + 10714062539980800 Pi - 45308459807170560 Pi / - 218193664957107840 Pi + 648299126654041009) / (16081747968 / 5 (15 Pi - 56) ) and in floating-point 359.4507029 127.4832976 4995.817765 6366.580531 50.85704738 - ----------- + ----------- + ----------- - ----------- m 2 3 4 m m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: 4 3 2 - 125 Pi (1563145902489600 Pi + 10714062539980800 Pi - 45308459807170560 Pi / - 218193664957107840 Pi + 648299126654041009) / (16081747968 / 5 1/2 4 (15 Pi - 56) ) + 625 Pi (5392677909343076352000 Pi 3 2 + 56698169647770503424000 Pi - 152582978834360819559360 Pi / - 1373693585534219078348415 Pi + 3538103163814237645570048) / ( / 6 6 763577475268608 (15 Pi - 56) m) - 125 (20446239630206164848353280000 Pi 5 4 - 106039425998753661505867776000 Pi + 355938104827757764726544239200 Pi 3 + 6697603408875001227313289197785 Pi 2 - 37084106526108697867268969253495 Pi + 5021903729419676952240493212912 Pi + 120682851895855331143457544601600) / 7 2 1/2 / (36255422103228776448 (15 Pi - 56) m ) - 625 Pi ( / 6 5 2358405675169835561434644480000 Pi + 3920177203996908098855939328000 Pi 4 - 289017799154695238656588438132800 Pi 3 + 412560453842127764330124169340145 Pi 2 + 4371033406789568940607852028242200 Pi - 10087605493631654663957603205264576 Pi / + 444726760390033372310166631874560) / (290043376825830211584 / 8 3 8 (15 Pi - 56) m ) + 125 (73328248650142425413650176000000 Pi 7 - 937121390622240977201656608000000 Pi 6 + 3829305649319381344635162512235000 Pi 5 + 12309977921240905142749291574998875 Pi 4 - 218181911639673887996501612017887525 Pi 3 + 851163805819371651393873378938445060 Pi 2 - 723097671962870411328389667582779680 Pi - 2090722358839856784624616735796438784 Pi / + 3252423015627133238411470799319859200) / (24170281402152517632 / 9 4 (15 Pi - 56) m ) and in Maple input form -125/16081747968*Pi*(1563145902489600*Pi^4+10714062539980800*Pi^3-\ 45308459807170560*Pi^2-218193664957107840*Pi+648299126654041009)/(15*Pi-56)^5+ 625/763577475268608*Pi^(1/2)*(5392677909343076352000*Pi^4+ 56698169647770503424000*Pi^3-152582978834360819559360*Pi^2-\ 1373693585534219078348415*Pi+3538103163814237645570048)/(15*Pi-56)^6/m-125/ 36255422103228776448*(20446239630206164848353280000*Pi^6-\ 106039425998753661505867776000*Pi^5+355938104827757764726544239200*Pi^4+ 6697603408875001227313289197785*Pi^3-37084106526108697867268969253495*Pi^2+ 5021903729419676952240493212912*Pi+120682851895855331143457544601600)/(15*Pi-56 )^7/m^2-625/290043376825830211584*Pi^(1/2)*(2358405675169835561434644480000*Pi^ 6+3920177203996908098855939328000*Pi^5-289017799154695238656588438132800*Pi^4+ 412560453842127764330124169340145*Pi^3+4371033406789568940607852028242200*Pi^2-\ 10087605493631654663957603205264576*Pi+444726760390033372310166631874560)/(15* Pi-56)^8/m^3+125/24170281402152517632*(73328248650142425413650176000000*Pi^8-\ 937121390622240977201656608000000*Pi^7+3829305649319381344635162512235000*Pi^6+ 12309977921240905142749291574998875*Pi^5-218181911639673887996501612017887525* Pi^4+851163805819371651393873378938445060*Pi^3-\ 723097671962870411328389667582779680*Pi^2-2090722358839856784624616735796438784 *Pi+3252423015627133238411470799319859200)/(15*Pi-56)^9/m^4 and in floating-point 359.4507029 127.4832976 4995.817765 6366.580531 50.85704738 - ----------- + ----------- + ----------- - ----------- m 2 3 4 m m m 3 The limit of the, 6, -th standarized moment is: , (13768017120000 Pi 2 / + 126895224456000 Pi - 330406854934275 Pi - 655973743198208) / ( / 3 815882496 (15 Pi - 56) ) and in floating-point 25.72396919 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 (13768017120000 Pi + 126895224456000 Pi - 330406854934275 Pi / 3 1/2 - 655973743198208) / (815882496 (15 Pi - 56) ) + 25 Pi / 2 / (4224608273340000 Pi - 127027389407114625 Pi + 356989558221533944) / ( / 4 4 26652161536 (15 Pi - 56) m) + (2087566990809300000 Pi 3 2 - 9524197040225135625 Pi - 20696077252423519875 Pi / + 599120316102537643960 Pi - 1585840036323738517504) / (26652161536 / 5 2 1/2 4 (15 Pi - 56) m ) - 5 Pi (1169931299322321900000 Pi 3 2 - 30570918372159869851875 Pi + 238894608833587768296600 Pi / - 655372745590269545836480 Pi + 535004070987780811286016) / ( / 6 3 6 639651876864 (15 Pi - 56) m ) - 5 (26952607565993859750000 Pi 5 4 - 315961098329644821703125 Pi + 690199013200674960331875 Pi 3 2 + 9081136471629703092175800 Pi - 73539826388797210966688640 Pi / + 207772124180864102829193728 Pi - 204951890820612185078628352) / ( / 7 4 319825938432 (15 Pi - 56) m ) and in Maple input form 1/815882496*(13768017120000*Pi^3+126895224456000*Pi^2-330406854934275*Pi-\ 655973743198208)/(15*Pi-56)^3+25/26652161536*Pi^(1/2)*(4224608273340000*Pi^2-\ 127027389407114625*Pi+356989558221533944)/(15*Pi-56)^4/m+1/26652161536*( 2087566990809300000*Pi^4-9524197040225135625*Pi^3-20696077252423519875*Pi^2+ 599120316102537643960*Pi-1585840036323738517504)/(15*Pi-56)^5/m^2-5/ 639651876864*Pi^(1/2)*(1169931299322321900000*Pi^4-30570918372159869851875*Pi^3 +238894608833587768296600*Pi^2-655372745590269545836480*Pi+ 535004070987780811286016)/(15*Pi-56)^6/m^3-5/319825938432*( 26952607565993859750000*Pi^6-315961098329644821703125*Pi^5+ 690199013200674960331875*Pi^4+9081136471629703092175800*Pi^3-\ 73539826388797210966688640*Pi^2+207772124180864102829193728*Pi-\ 204951890820612185078628352)/(15*Pi-56)^7/m^4 and in floating-point 102.7319667 87.14693687 1225.571395 45.02021238 25.72396919 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m This ends this exciting article that took, 105.765, seconds to produce ------------------------------------------ For #231 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](t, q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: G[n](t, q) = n ----- \ ((k - 1) (n - k)) (k - 1) (n - k) ) t q G[k - 1](t, q t ) G[n - k](t, q) / ----- k = 1 Note that what we REALLY care about is the statistic carried by the variable, t, and the remaining variable , q serves as a CATALYTIC variable, to enable the recurrence. At the end of the day we are really only interested in the random variable defined on the n-th set of our family whose weight-enumerator is: G(t, 1)(n) 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 1/2 5 4 1/2 3 2 1/2 Pi m 3 m 41 Pi m 5 m 593 Pi m -------- - ---- + ----------- - ---- + ----------- 8 4 64 4 1024 and in Maple input form 1/8*Pi^(1/2)*m^5-3/4*m^4+41/64*Pi^(1/2)*m^3-5/4*m^2+593/1024*Pi^(1/2)*m and in floating-point: .2215567314*m^5-.7500000000*m^4+1.135478248*m^3-1.250000000*m^2+1.026430795*m The asymoptotics of the Variance to order, 4, is: 1/2 9 1/2 7 / Pi 43 \ 10 Pi m / 41 Pi 293\ 8 27 Pi m |- ---- + ---| m - -------- + |- ----- + ---| m - ----------- \ 64 840/ 384 \ 256 560/ 1024 / 1137 Pi 187\ 6 + |- ------- + ---| m \ 2048 105/ and in Maple input form (-1/64*Pi+43/840)*m^10-1/384*Pi^(1/2)*m^9+(-41/256*Pi+293/560)*m^8-27/1024*Pi^( 1/2)*m^7+(-1137/2048*Pi+187/105)*m^6 and in floating-point: .210309097e-2*m^10-.4615765238e-2*m^9+.200685872e-1*m^8-.4673462302e-1*m^7+.\ 36816225e-1*m^6 The asympotic skewness squared is, 0.5834556795 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 2625 Pi (7056 Pi - 44184 Pi + 69169) - ------------------------------------- 3 4 (105 Pi - 344) 1/2 2 11025 Pi (7055580 Pi - 44282813 Pi + 69482496) + -------------------------------------------------- + 315 ( 4 88 (105 Pi - 344) m 4 3 2 96743493000 Pi - 1044025388175 Pi + 3435283084870 Pi - 2671310121600 Pi / 5 2 1/2 - 2565054922752) / (1936 (105 Pi - 344) m ) - 35 Pi ( / 4 3 2 1305498767265000 Pi - 16114274294146875 Pi + 73813817244709500 Pi / 6 3 - 148575609112969920 Pi + 110727052970688512) / (7744 (105 Pi - 344) m / 6 5 ) + 105 (675415350051750000 Pi - 10819602320614250625 Pi 4 3 + 69510250687419486000 Pi - 224374595578878685500 Pi 2 + 366242162297221378560 Pi - 249222534232551092224 Pi / 7 4 + 16055150080168034304) / (30976 (105 Pi - 344) m ) / and in Maple input form -2625/4*Pi*(7056*Pi^2-44184*Pi+69169)/(105*Pi-344)^3+11025/88*Pi^(1/2)*(7055580 *Pi^2-44282813*Pi+69482496)/(105*Pi-344)^4/m+315/1936*(96743493000*Pi^4-\ 1044025388175*Pi^3+3435283084870*Pi^2-2671310121600*Pi-2565054922752)/(105*Pi-\ 344)^5/m^2-35/7744*Pi^(1/2)*(1305498767265000*Pi^4-16114274294146875*Pi^3+ 73813817244709500*Pi^2-148575609112969920*Pi+110727052970688512)/(105*Pi-344)^6 /m^3+105/30976*(675415350051750000*Pi^6-10819602320614250625*Pi^5+ 69510250687419486000*Pi^4-224374595578878685500*Pi^3+366242162297221378560*Pi^2 -249222534232551092224*Pi+16055150080168034304)/(105*Pi-344)^7/m^4 and in floating-point 1.561718554 5.064894591 12.56676988 47.85967770 0.5834556795 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 7 (34459425 Pi - 61917570 Pi - 146378816) - ------------------------------------------ 2 7293 (105 Pi - 344) and in floating-point 3.833366074 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 7 (34459425 Pi - 61917570 Pi - 146378816) - ------------------------------------------ 2 7293 (105 Pi - 344) 1/2 245 Pi (1218352005 Pi - 3826472824) + -------------------------------------- + 7 3 58344 (105 Pi - 344) m 3 2 (1249498750500 Pi - 9935937407025 Pi + 21490555412280 Pi - 8195716317184) / 4 2 1/2 3 / (58344 (105 Pi - 344) m ) - 35 Pi (43578187687125 Pi / 2 / - 197172892687050 Pi - 219524942512400 Pi + 1284579445901824) / (466752 / 5 3 5 4 (105 Pi - 344) m ) + 7 (363474332048953125 Pi - 5076542822152016250 Pi 3 2 + 27638852289600092625 Pi - 72502587473422051400 Pi / + 89765897995575463680 Pi - 40142800830989336576) / (116688 / 6 4 (105 Pi - 344) m ) and in Maple input form -7/7293*(34459425*Pi^2-61917570*Pi-146378816)/(105*Pi-344)^2+245/58344*Pi^(1/2) *(1218352005*Pi-3826472824)/(105*Pi-344)^3/m+7/58344*(1249498750500*Pi^3-\ 9935937407025*Pi^2+21490555412280*Pi-8195716317184)/(105*Pi-344)^4/m^2-35/ 466752*Pi^(1/2)*(43578187687125*Pi^3-197172892687050*Pi^2-219524942512400*Pi+ 1284579445901824)/(105*Pi-344)^5/m^3+7/116688*(363474332048953125*Pi^5-\ 5076542822152016250*Pi^4+27638852289600092625*Pi^3-72502587473422051400*Pi^2+ 89765897995575463680*Pi-40142800830989336576)/(105*Pi-344)^6/m^4 and in floating-point 2.881623863 7.858113471 23.66616541 63.08874316 3.833366074 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The square of the limit of the, 5, -th standarized moment is: 4 3 - 3215625 Pi (98380929445134336 Pi + 142886588003647488 Pi 2 - 2337384361183432704 Pi - 1735062069683205120 Pi + 14506345638906499225) / 5 / (1549208387584 (105 Pi - 344) ) / and in floating-point 298.1616958 605.8942695 1411.664834 352618.7481 74.13559940 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: 4 3 - 3215625 Pi (98380929445134336 Pi + 142886588003647488 Pi 2 - 2337384361183432704 Pi - 1735062069683205120 Pi + 14506345638906499225) / 5 1/2 / (1549208387584 (105 Pi - 344) ) + 30625 Pi ( / 4 3 242159207749299929088000 Pi + 1060779813724217231124480 Pi 2 - 7468949599411250473705728 Pi - 14500765746248548767196617 Pi / 6 + 62791728528458980545003520) / (176609756184576 (105 Pi - 344) m) + / 6 875 (239589787595645497847316480000 Pi 5 - 1608744579774391474708220928000 Pi 4 - 1207540739393447243105532556800 Pi 3 + 23127444557795191330861969236300 Pi 2 - 33613398260030942723307228279975 Pi + 19798403586251214060847580896320 Pi - 67949593745666829844585293807616) / 7 2 1/2 / (60400536615124992 (105 Pi - 344) m ) + 30625 Pi ( / 6 5 9070177808112570094722416640000 Pi - 125678674343741423139353454336000 Pi 4 + 483206054008098681360120798086400 Pi 3 + 629026316335593050908706052448635 Pi 2 - 8645942595818188177175705855761386 Pi + 20326979901327677753733273231556032 Pi / - 15359306732084680946669551584542720) / (241602146460499968 / 8 3 8 (105 Pi - 344) m ) + 875 (46463825581914311944859566080000000 Pi 7 - 605775459232671852403872783590400000 Pi 6 + 2377193637128657793776752882037760000 Pi 5 + 1611214558388086252255092634075047000 Pi 4 - 37166798311878743454677650442304297075 Pi 3 + 103189753947546687569958681101961384150 Pi 2 - 102643399509433453512040667014318195200 Pi + 3701270290505252229872506540784814080 Pi / + 32550264574746428355907640222984175616) / (80534048820166656 / 9 4 (105 Pi - 344) m ) and in Maple input form -3215625/1549208387584*Pi*(98380929445134336*Pi^4+142886588003647488*Pi^3-\ 2337384361183432704*Pi^2-1735062069683205120*Pi+14506345638906499225)/(105*Pi-\ 344)^5+30625/176609756184576*Pi^(1/2)*(242159207749299929088000*Pi^4+ 1060779813724217231124480*Pi^3-7468949599411250473705728*Pi^2-\ 14500765746248548767196617*Pi+62791728528458980545003520)/(105*Pi-344)^6/m+875/ 60400536615124992*(239589787595645497847316480000*Pi^6-\ 1608744579774391474708220928000*Pi^5-1207540739393447243105532556800*Pi^4+ 23127444557795191330861969236300*Pi^3-33613398260030942723307228279975*Pi^2+ 19798403586251214060847580896320*Pi-67949593745666829844585293807616)/(105*Pi-\ 344)^7/m^2+30625/241602146460499968*Pi^(1/2)*(9070177808112570094722416640000* Pi^6-125678674343741423139353454336000*Pi^5+483206054008098681360120798086400* Pi^4+629026316335593050908706052448635*Pi^3-8645942595818188177175705855761386* Pi^2+20326979901327677753733273231556032*Pi-15359306732084680946669551584542720 )/(105*Pi-344)^8/m^3+875/80534048820166656*(46463825581914311944859566080000000 *Pi^8-605775459232671852403872783590400000*Pi^7+ 2377193637128657793776752882037760000*Pi^6+ 1611214558388086252255092634075047000*Pi^5-\ 37166798311878743454677650442304297075*Pi^4+ 103189753947546687569958681101961384150*Pi^3-\ 102643399509433453512040667014318195200*Pi^2+ 3701270290505252229872506540784814080*Pi+32550264574746428355907640222984175616 )/(105*Pi-344)^9/m^4 and in floating-point 298.1616958 605.8942695 1411.664834 352618.7481 74.13559940 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m 3 The limit of the, 6, -th standarized moment is: , 7 (2023898516640000 Pi 2 / + 8982254369088000 Pi - 40589954834418675 Pi - 23921383426883584) / ( / 3 2447647488 (105 Pi - 344) ) and in floating-point 34.07443403 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 7 (2023898516640000 Pi + 8982254369088000 Pi - 40589954834418675 Pi / 3 1/2 - 23921383426883584) / (2447647488 (105 Pi - 344) ) + 35 Pi / 2 / (72878745676464000 Pi - 1577250464279289975 Pi + 4235636779009000856) / / 4 4 (3263529984 (105 Pi - 344) m) - 175 (2935462408534656000 Pi 3 2 - 17986242410143591500 Pi + 16732504513583562375 Pi / + 9247496171532253432 Pi + 77546949457609752576) / (6527059968 / 5 2 1/2 4 (105 Pi - 344) m ) - 35 Pi (120523744056458761200000 Pi 3 2 - 1630520698368147286640625 Pi + 8130458997751634598341700 Pi / - 17650925937684492615020320 Pi + 14023864492856031878235648) / ( / 6 3 6 78324719616 (105 Pi - 344) m ) - 7 (128087191496188524600000000 Pi 5 4 - 1560855423823430487570562500 Pi + 6274172186496937093188628125 Pi 3 2 - 4553516736863838134824608000 Pi - 31696890692959403301467590800 Pi / + 80800760652329245004792394240 Pi - 56470269479859404781905772544) / ( / 7 4 78324719616 (105 Pi - 344) m ) and in Maple input form 7/2447647488*(2023898516640000*Pi^3+8982254369088000*Pi^2-40589954834418675*Pi-\ 23921383426883584)/(105*Pi-344)^3+35/3263529984*Pi^(1/2)*(72878745676464000*Pi^ 2-1577250464279289975*Pi+4235636779009000856)/(105*Pi-344)^4/m-175/6527059968*( 2935462408534656000*Pi^4-17986242410143591500*Pi^3+16732504513583562375*Pi^2+ 9247496171532253432*Pi+77546949457609752576)/(105*Pi-344)^5/m^2-35/78324719616* Pi^(1/2)*(120523744056458761200000*Pi^4-1630520698368147286640625*Pi^3+ 8130458997751634598341700*Pi^2-17650925937684492615020320*Pi+ 14023864492856031878235648)/(105*Pi-344)^6/m^3-7/78324719616*( 128087191496188524600000000*Pi^6-1560855423823430487570562500*Pi^5+ 6274172186496937093188628125*Pi^4-4553516736863838134824608000*Pi^3-\ 31696890692959403301467590800*Pi^2+80800760652329245004792394240*Pi-\ 56470269479859404781905772544)/(105*Pi-344)^7/m^4 and in floating-point 74.95197040 174.4248592 833.9540184 1174.544806 34.07443403 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m This ends this exciting article that took, 457.923, seconds to produce ------------------------------------------ For #312 in Av(132): Explicit (Rigorously-Derived) Asymptotics of order, 4, for The Expectation, Variance, Skewness, Kurtosis and all the Standarized Moments up to the, 6, -th of a Certain Combinatorial Statistic Defined on an Infinite Sequence of Combinatorial Sets Whose Generating Function Satisfies a Non-Linear (Quadratic) Functional Recurrence Equation By Shalosh B. Ekhad Theorem: let, G[n](t, q), be a sequence of polynomial generating functions satisfying the Quadratic Functional Equation: n ----- \ (k - 1) k G[n](t, q) = ) q G[k - 1](t, q) G[n - k](t, t q) / ----- k = 1 Note that what we REALLY care about is the statistic carried by the variable, t, and the remaining variable , q serves as a CATALYTIC variable, to enable the recurrence. At the end of the day we are really only interested in the random variable defined on the n-th set of our family whose weight-enumerator is: G(t, 1)(n) 1/2 In this article, m = n The asymoptotics of the Expectation to order, 4, is: 1/2 5 4 1/2 3 2 1/2 Pi m 3 m 41 Pi m 5 m 593 Pi m -------- - ---- + ----------- - ---- + ----------- 8 4 64 4 1024 and in Maple input form 1/8*Pi^(1/2)*m^5-3/4*m^4+41/64*Pi^(1/2)*m^3-5/4*m^2+593/1024*Pi^(1/2)*m and in floating-point: .2215567314*m^5-.7500000000*m^4+1.135478248*m^3-1.250000000*m^2+1.026430795*m The asymoptotics of the Variance to order, 4, is: 1/2 9 1/2 7 / Pi 43 \ 10 Pi m / 41 Pi 293\ 8 27 Pi m |- ---- + ---| m - -------- + |- ----- + ---| m - ----------- \ 64 840/ 384 \ 256 560/ 1024 / 1137 Pi 187\ 6 + |- ------- + ---| m \ 2048 105/ and in Maple input form (-1/64*Pi+43/840)*m^10-1/384*Pi^(1/2)*m^9+(-41/256*Pi+293/560)*m^8-27/1024*Pi^( 1/2)*m^7+(-1137/2048*Pi+187/105)*m^6 and in floating-point: .210309097e-2*m^10-.4615765238e-2*m^9+.200685872e-1*m^8-.4673462302e-1*m^7+.\ 36816225e-1*m^6 The asympotic skewness squared is, 0.5834556795 More precisely, the asymoptotics of the SQUARE of the, 3, -th standarized moment is: 2 2625 Pi (7056 Pi - 44184 Pi + 69169) - ------------------------------------- 3 4 (105 Pi - 344) 1/2 2 11025 Pi (7055580 Pi - 44282813 Pi + 69482496) + -------------------------------------------------- + 315 ( 4 88 (105 Pi - 344) m 4 3 2 96743493000 Pi - 1044025388175 Pi + 3435283084870 Pi - 2671310121600 Pi / 5 2 1/2 - 2565054922752) / (1936 (105 Pi - 344) m ) - 35 Pi ( / 4 3 2 1305498767265000 Pi - 16114274294146875 Pi + 73813817244709500 Pi / 6 3 - 148575609112969920 Pi + 110727052970688512) / (7744 (105 Pi - 344) m / 6 5 ) + 105 (675415350051750000 Pi - 10819602320614250625 Pi 4 3 + 69510250687419486000 Pi - 224374595578878685500 Pi 2 + 366242162297221378560 Pi - 249222534232551092224 Pi / 7 4 + 16055150080168034304) / (30976 (105 Pi - 344) m ) / and in Maple input form -2625/4*Pi*(7056*Pi^2-44184*Pi+69169)/(105*Pi-344)^3+11025/88*Pi^(1/2)*(7055580 *Pi^2-44282813*Pi+69482496)/(105*Pi-344)^4/m+315/1936*(96743493000*Pi^4-\ 1044025388175*Pi^3+3435283084870*Pi^2-2671310121600*Pi-2565054922752)/(105*Pi-\ 344)^5/m^2-35/7744*Pi^(1/2)*(1305498767265000*Pi^4-16114274294146875*Pi^3+ 73813817244709500*Pi^2-148575609112969920*Pi+110727052970688512)/(105*Pi-344)^6 /m^3+105/30976*(675415350051750000*Pi^6-10819602320614250625*Pi^5+ 69510250687419486000*Pi^4-224374595578878685500*Pi^3+366242162297221378560*Pi^2 -249222534232551092224*Pi+16055150080168034304)/(105*Pi-344)^7/m^4 and in floating-point 1.561718554 5.064894591 12.56676988 47.85967770 0.5834556795 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The limit of the kurtosis (aka 4th standarized moment) is: , 2 7 (34459425 Pi - 61917570 Pi - 146378816) - ------------------------------------------ 2 7293 (105 Pi - 344) and in floating-point 3.833366074 more precisely,to order , 4 the asymoptotics of the, 4, -th standarized moment is: 2 7 (34459425 Pi - 61917570 Pi - 146378816) - ------------------------------------------ 2 7293 (105 Pi - 344) 1/2 245 Pi (1218352005 Pi - 3826472824) + -------------------------------------- + 7 3 58344 (105 Pi - 344) m 3 2 (1249498750500 Pi - 9935937407025 Pi + 21490555412280 Pi - 8195716317184) / 4 2 1/2 3 / (58344 (105 Pi - 344) m ) - 35 Pi (43578187687125 Pi / 2 / - 197172892687050 Pi - 219524942512400 Pi + 1284579445901824) / (466752 / 5 3 5 4 (105 Pi - 344) m ) + 7 (363474332048953125 Pi - 5076542822152016250 Pi 3 2 + 27638852289600092625 Pi - 72502587473422051400 Pi / + 89765897995575463680 Pi - 40142800830989336576) / (116688 / 6 4 (105 Pi - 344) m ) and in Maple input form -7/7293*(34459425*Pi^2-61917570*Pi-146378816)/(105*Pi-344)^2+245/58344*Pi^(1/2) *(1218352005*Pi-3826472824)/(105*Pi-344)^3/m+7/58344*(1249498750500*Pi^3-\ 9935937407025*Pi^2+21490555412280*Pi-8195716317184)/(105*Pi-344)^4/m^2-35/ 466752*Pi^(1/2)*(43578187687125*Pi^3-197172892687050*Pi^2-219524942512400*Pi+ 1284579445901824)/(105*Pi-344)^5/m^3+7/116688*(363474332048953125*Pi^5-\ 5076542822152016250*Pi^4+27638852289600092625*Pi^3-72502587473422051400*Pi^2+ 89765897995575463680*Pi-40142800830989336576)/(105*Pi-344)^6/m^4 and in floating-point 2.881623863 7.858113471 23.66616541 63.08874316 3.833366074 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m The square of the limit of the, 5, -th standarized moment is: 4 3 - 3215625 Pi (98380929445134336 Pi + 142886588003647488 Pi 2 - 2337384361183432704 Pi - 1735062069683205120 Pi + 14506345638906499225) / 5 / (1549208387584 (105 Pi - 344) ) / and in floating-point 298.1616958 605.8942695 1411.664834 352618.7481 74.13559940 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m More precisely, the asymoptotics of the SQUARE of the, 5, -th standarized moment is: 4 3 - 3215625 Pi (98380929445134336 Pi + 142886588003647488 Pi 2 - 2337384361183432704 Pi - 1735062069683205120 Pi + 14506345638906499225) / 5 1/2 / (1549208387584 (105 Pi - 344) ) + 30625 Pi ( / 4 3 242159207749299929088000 Pi + 1060779813724217231124480 Pi 2 - 7468949599411250473705728 Pi - 14500765746248548767196617 Pi / 6 + 62791728528458980545003520) / (176609756184576 (105 Pi - 344) m) + / 6 875 (239589787595645497847316480000 Pi 5 - 1608744579774391474708220928000 Pi 4 - 1207540739393447243105532556800 Pi 3 + 23127444557795191330861969236300 Pi 2 - 33613398260030942723307228279975 Pi + 19798403586251214060847580896320 Pi - 67949593745666829844585293807616) / 7 2 1/2 / (60400536615124992 (105 Pi - 344) m ) + 30625 Pi ( / 6 5 9070177808112570094722416640000 Pi - 125678674343741423139353454336000 Pi 4 + 483206054008098681360120798086400 Pi 3 + 629026316335593050908706052448635 Pi 2 - 8645942595818188177175705855761386 Pi + 20326979901327677753733273231556032 Pi / - 15359306732084680946669551584542720) / (241602146460499968 / 8 3 8 (105 Pi - 344) m ) + 875 (46463825581914311944859566080000000 Pi 7 - 605775459232671852403872783590400000 Pi 6 + 2377193637128657793776752882037760000 Pi 5 + 1611214558388086252255092634075047000 Pi 4 - 37166798311878743454677650442304297075 Pi 3 + 103189753947546687569958681101961384150 Pi 2 - 102643399509433453512040667014318195200 Pi + 3701270290505252229872506540784814080 Pi / + 32550264574746428355907640222984175616) / (80534048820166656 / 9 4 (105 Pi - 344) m ) and in Maple input form -3215625/1549208387584*Pi*(98380929445134336*Pi^4+142886588003647488*Pi^3-\ 2337384361183432704*Pi^2-1735062069683205120*Pi+14506345638906499225)/(105*Pi-\ 344)^5+30625/176609756184576*Pi^(1/2)*(242159207749299929088000*Pi^4+ 1060779813724217231124480*Pi^3-7468949599411250473705728*Pi^2-\ 14500765746248548767196617*Pi+62791728528458980545003520)/(105*Pi-344)^6/m+875/ 60400536615124992*(239589787595645497847316480000*Pi^6-\ 1608744579774391474708220928000*Pi^5-1207540739393447243105532556800*Pi^4+ 23127444557795191330861969236300*Pi^3-33613398260030942723307228279975*Pi^2+ 19798403586251214060847580896320*Pi-67949593745666829844585293807616)/(105*Pi-\ 344)^7/m^2+30625/241602146460499968*Pi^(1/2)*(9070177808112570094722416640000* Pi^6-125678674343741423139353454336000*Pi^5+483206054008098681360120798086400* Pi^4+629026316335593050908706052448635*Pi^3-8645942595818188177175705855761386* Pi^2+20326979901327677753733273231556032*Pi-15359306732084680946669551584542720 )/(105*Pi-344)^8/m^3+875/80534048820166656*(46463825581914311944859566080000000 *Pi^8-605775459232671852403872783590400000*Pi^7+ 2377193637128657793776752882037760000*Pi^6+ 1611214558388086252255092634075047000*Pi^5-\ 37166798311878743454677650442304297075*Pi^4+ 103189753947546687569958681101961384150*Pi^3-\ 102643399509433453512040667014318195200*Pi^2+ 3701270290505252229872506540784814080*Pi+32550264574746428355907640222984175616 )/(105*Pi-344)^9/m^4 and in floating-point 298.1616958 605.8942695 1411.664834 352618.7481 74.13559940 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m 3 The limit of the, 6, -th standarized moment is: , 7 (2023898516640000 Pi 2 / + 8982254369088000 Pi - 40589954834418675 Pi - 23921383426883584) / ( / 3 2447647488 (105 Pi - 344) ) and in floating-point 34.07443403 more precisely,to order , 4 the asymoptotics of the, 6, -th standarized moment is: 3 2 7 (2023898516640000 Pi + 8982254369088000 Pi - 40589954834418675 Pi / 3 1/2 - 23921383426883584) / (2447647488 (105 Pi - 344) ) + 35 Pi / 2 / (72878745676464000 Pi - 1577250464279289975 Pi + 4235636779009000856) / / 4 4 (3263529984 (105 Pi - 344) m) - 175 (2935462408534656000 Pi 3 2 - 17986242410143591500 Pi + 16732504513583562375 Pi / + 9247496171532253432 Pi + 77546949457609752576) / (6527059968 / 5 2 1/2 4 (105 Pi - 344) m ) - 35 Pi (120523744056458761200000 Pi 3 2 - 1630520698368147286640625 Pi + 8130458997751634598341700 Pi / - 17650925937684492615020320 Pi + 14023864492856031878235648) / ( / 6 3 6 78324719616 (105 Pi - 344) m ) - 7 (128087191496188524600000000 Pi 5 4 - 1560855423823430487570562500 Pi + 6274172186496937093188628125 Pi 3 2 - 4553516736863838134824608000 Pi - 31696890692959403301467590800 Pi / + 80800760652329245004792394240 Pi - 56470269479859404781905772544) / ( / 7 4 78324719616 (105 Pi - 344) m ) and in Maple input form 7/2447647488*(2023898516640000*Pi^3+8982254369088000*Pi^2-40589954834418675*Pi-\ 23921383426883584)/(105*Pi-344)^3+35/3263529984*Pi^(1/2)*(72878745676464000*Pi^ 2-1577250464279289975*Pi+4235636779009000856)/(105*Pi-344)^4/m-175/6527059968*( 2935462408534656000*Pi^4-17986242410143591500*Pi^3+16732504513583562375*Pi^2+ 9247496171532253432*Pi+77546949457609752576)/(105*Pi-344)^5/m^2-35/78324719616* Pi^(1/2)*(120523744056458761200000*Pi^4-1630520698368147286640625*Pi^3+ 8130458997751634598341700*Pi^2-17650925937684492615020320*Pi+ 14023864492856031878235648)/(105*Pi-344)^6/m^3-7/78324719616*( 128087191496188524600000000*Pi^6-1560855423823430487570562500*Pi^5+ 6274172186496937093188628125*Pi^4-4553516736863838134824608000*Pi^3-\ 31696890692959403301467590800*Pi^2+80800760652329245004792394240*Pi-\ 56470269479859404781905772544)/(105*Pi-344)^7/m^4 and in floating-point 74.95197040 174.4248592 833.9540184 1174.544806 34.07443403 - ----------- - ----------- + ----------- + ----------- m 2 3 4 m m m This ends this exciting article that took, 96.838, seconds to produce ------------------------------------------ This took, 2377.579, seconds altogether.