Explicit (and Asymptotic) Expressions for the Expectation, Variance, and all\ Moments (about the mean) up to the, 12, -th of the random variable: Total Height defined on the set of Rooted Labeled Tr\ ees with n vertices By Shalosh B. Ekhad Arthur Cayley famously proved that the number of labeled rooted trees on n v\ (n - 1) ertices, let's call it r(n), equals, n , . One of the many ways of proving it is by noting that the exponential generat\ ing function, let's call it, R(x), defined by infinity ----- n \ r(n) x R(x) = ) ------- / n! ----- n = 0 satisfies the Functional Equation, that follows from standard generatingfunc\ tionlogy R(x) = x exp(R(x)) and it follows from the famous Lagrange Inversion Formula (e.g. http://www.m\ ath.rutgers.edu/~zeilberg/mamarim/mamarimPDF/lag.pdf) r(n) (n - 1) that , ----, equals 1/n times the coefficient of , x , n! n in the formal power series, exp(x) hence r(n) equals (n-1)! times n^(n-1)/(n-1)!= n^(n-1) , . In a delightful article (and very historically significant, since that's how\ the OEIS started!) [Available on-line from http://neilsloane.com/doc/riordan-enum-trees-by-hei\ ght.pdf], entitled "The Enumeration of Rooted Trees by Total Height" J. Australian Math. Soc., 10 (1969), pp. 278-282, John Riordan and Neil Sloane considered the polynomials, J[n](y), defined as the weight-enumerators of the set of rooted labeled trees on n vertices, according to the weight, "total height", where the height of \ a vertex is its distance to the root and the total height of a rooted labeled tree is the sum of the heights (dis\ tance to the root) of the vertices Let J(x,y) be the exponential generating function of the sequence J[n](y) infinity ----- n \ J[n](y) x J(x, y) = ) ----------, . / n! ----- n = 0 The same generatingfunctionology argument that lead to the functional equati\ on for R(x), leads to the functional equation J(x, y) = x exp(J(x y, y)) Alas, it does not seem to be possible to get a closed-form expression for th\ e polynomials J[n](y) (for symbolic n) . For the record, here are the first 12 members of the sequence of polynomials, J[n](y) 3 2 6 5 4 3 [1, 2 y, 6 y + 3 y , 24 y + 12 y + 24 y + 4 y , 10 9 8 7 6 5 4 15 120 y + 60 y + 120 y + 140 y + 120 y + 60 y + 5 y , 720 y 14 13 12 11 10 9 8 + 360 y + 720 y + 840 y + 1440 y + 720 y + 1470 y + 840 y 7 6 5 21 20 19 18 + 540 y + 120 y + 6 y , 5040 y + 2520 y + 5040 y + 5880 y 17 16 15 14 13 12 + 10080 y + 10080 y + 12810 y + 10920 y + 14700 y + 13440 y 11 10 9 8 7 6 28 + 11382 y + 9240 y + 4620 y + 1680 y + 210 y + 7 y , 40320 y 27 26 25 24 23 22 + 20160 y + 40320 y + 47040 y + 80640 y + 80640 y + 142800 y 21 20 19 18 17 + 107520 y + 157920 y + 154560 y + 212016 y + 154560 y 16 15 14 13 12 + 210000 y + 168000 y + 186480 y + 121016 y + 99456 y 11 10 9 8 7 36 35 + 49560 y + 19600 y + 4200 y + 336 y + 8 y , 362880 y + 181440 y 34 33 32 31 30 + 362880 y + 423360 y + 725760 y + 725760 y + 1285200 y 29 28 27 26 25 + 1330560 y + 1602720 y + 1753920 y + 2331504 y + 2116800 y 24 23 22 21 20 + 2978640 y + 2797200 y + 2918160 y + 3115224 y + 3344544 y 19 18 17 16 15 + 2989224 y + 3018960 y + 2736720 y + 2210544 y + 1718712 y 14 13 12 11 10 9 + 1089648 y + 613872 y + 236880 y + 66024 y + 9072 y + 504 y 8 45 44 43 42 + 9 y , 3628800 y + 1814400 y + 3628800 y + 4233600 y 41 40 39 38 37 + 7257600 y + 7257600 y + 12852000 y + 13305600 y + 19656000 y 36 35 34 33 32 + 19353600 y + 26943840 y + 25401600 y + 37044000 y + 35229600 y 31 30 29 28 27 + 45662400 y + 44457840 y + 52194240 y + 49850640 y + 62274240 y 26 25 24 23 22 + 55792800 y + 65651040 y + 56877120 y + 63665280 y + 56190960 y 21 20 19 18 17 + 58474080 y + 46186560 y + 43384320 y + 31404240 y + 23899770 y 16 15 14 13 12 + 14208480 y + 7849800 y + 3195360 y + 968940 y + 186480 y 11 10 9 55 54 53 + 17640 y + 720 y + 10 y , 39916800 y + 19958400 y + 39916800 y 52 51 50 49 + 46569600 y + 79833600 y + 79833600 y + 141372000 y 48 47 46 45 + 146361600 y + 216216000 y + 252806400 y + 316340640 y 44 43 42 41 + 319334400 y + 454053600 y + 467359200 y + 582120000 y 40 39 38 37 + 630408240 y + 760415040 y + 764573040 y + 927843840 y 36 35 34 33 + 936714240 y + 1058127840 y + 1112965920 y + 1229215680 y 32 31 30 29 + 1206041760 y + 1328508720 y + 1313373600 y + 1375189200 y 28 27 26 25 + 1342424160 y + 1351461870 y + 1280117520 y + 1224816120 y 24 23 22 21 + 1137517920 y + 1007053740 y + 843575040 y + 671877360 y 20 19 18 17 + 512661600 y + 339611690 y + 209135520 y + 107086320 y 16 15 14 13 12 + 46163040 y + 14677740 y + 3381840 y + 460680 y + 31680 y 11 10 66 65 64 + 990 y + 11 y , 479001600 y + 239500800 y + 479001600 y 63 62 61 60 + 558835200 y + 958003200 y + 958003200 y + 1696464000 y 59 58 57 56 + 1756339200 y + 2594592000 y + 3033676800 y + 4275089280 y 55 54 53 52 + 4071513600 y + 5927644800 y + 6167145600 y + 7943443200 y 51 50 49 48 + 8522902080 y + 10821444480 y + 10931215680 y + 14207719680 y 47 46 45 44 + 14274247680 y + 16852872960 y + 17506938240 y + 20877816960 y 43 42 41 40 + 20559813120 y + 24624008640 y + 24712490880 y + 27463423680 y 39 38 37 36 + 27180014400 y + 30908255400 y + 29613703680 y + 33475986720 y 35 34 33 32 + 32135114880 y + 34110069840 y + 32542836480 y + 34656763680 y 31 30 29 28 + 31449876480 y + 32748575640 y + 29481408000 y + 28787057640 y 27 26 25 24 + 24917365440 y + 23105221920 y + 18568630080 y + 15617432160 y 23 22 21 20 + 11524360320 y + 8479171800 y + 5385853572 y + 3210147600 y 19 18 17 16 + 1614319740 y + 697173840 y + 235077480 y + 59457024 y 15 14 13 12 11 + 10268280 y + 1025640 y + 53460 y + 1320 y + 12 y ] and in Maple notation [1, 2*y, 6*y^3+3*y^2, 24*y^6+12*y^5+24*y^4+4*y^3, 120*y^10+60*y^9+120*y^8+140*y ^7+120*y^6+60*y^5+5*y^4, 720*y^15+360*y^14+720*y^13+840*y^12+1440*y^11+720*y^10 +1470*y^9+840*y^8+540*y^7+120*y^6+6*y^5, 5040*y^21+2520*y^20+5040*y^19+5880*y^ 18+10080*y^17+10080*y^16+12810*y^15+10920*y^14+14700*y^13+13440*y^12+11382*y^11 +9240*y^10+4620*y^9+1680*y^8+210*y^7+7*y^6, 40320*y^28+20160*y^27+40320*y^26+ 47040*y^25+80640*y^24+80640*y^23+142800*y^22+107520*y^21+157920*y^20+154560*y^ 19+212016*y^18+154560*y^17+210000*y^16+168000*y^15+186480*y^14+121016*y^13+ 99456*y^12+49560*y^11+19600*y^10+4200*y^9+336*y^8+8*y^7, 362880*y^36+181440*y^ 35+362880*y^34+423360*y^33+725760*y^32+725760*y^31+1285200*y^30+1330560*y^29+ 1602720*y^28+1753920*y^27+2331504*y^26+2116800*y^25+2978640*y^24+2797200*y^23+ 2918160*y^22+3115224*y^21+3344544*y^20+2989224*y^19+3018960*y^18+2736720*y^17+ 2210544*y^16+1718712*y^15+1089648*y^14+613872*y^13+236880*y^12+66024*y^11+9072* y^10+504*y^9+9*y^8, 3628800*y^45+1814400*y^44+3628800*y^43+4233600*y^42+7257600 *y^41+7257600*y^40+12852000*y^39+13305600*y^38+19656000*y^37+19353600*y^36+ 26943840*y^35+25401600*y^34+37044000*y^33+35229600*y^32+45662400*y^31+44457840* y^30+52194240*y^29+49850640*y^28+62274240*y^27+55792800*y^26+65651040*y^25+ 56877120*y^24+63665280*y^23+56190960*y^22+58474080*y^21+46186560*y^20+43384320* y^19+31404240*y^18+23899770*y^17+14208480*y^16+7849800*y^15+3195360*y^14+968940 *y^13+186480*y^12+17640*y^11+720*y^10+10*y^9, 39916800*y^55+19958400*y^54+ 39916800*y^53+46569600*y^52+79833600*y^51+79833600*y^50+141372000*y^49+ 146361600*y^48+216216000*y^47+252806400*y^46+316340640*y^45+319334400*y^44+ 454053600*y^43+467359200*y^42+582120000*y^41+630408240*y^40+760415040*y^39+ 764573040*y^38+927843840*y^37+936714240*y^36+1058127840*y^35+1112965920*y^34+ 1229215680*y^33+1206041760*y^32+1328508720*y^31+1313373600*y^30+1375189200*y^29 +1342424160*y^28+1351461870*y^27+1280117520*y^26+1224816120*y^25+1137517920*y^ 24+1007053740*y^23+843575040*y^22+671877360*y^21+512661600*y^20+339611690*y^19+ 209135520*y^18+107086320*y^17+46163040*y^16+14677740*y^15+3381840*y^14+460680*y ^13+31680*y^12+990*y^11+11*y^10, 479001600*y^66+239500800*y^65+479001600*y^64+ 558835200*y^63+958003200*y^62+958003200*y^61+1696464000*y^60+1756339200*y^59+ 2594592000*y^58+3033676800*y^57+4275089280*y^56+4071513600*y^55+5927644800*y^54 +6167145600*y^53+7943443200*y^52+8522902080*y^51+10821444480*y^50+10931215680*y ^49+14207719680*y^48+14274247680*y^47+16852872960*y^46+17506938240*y^45+ 20877816960*y^44+20559813120*y^43+24624008640*y^42+24712490880*y^41+27463423680 *y^40+27180014400*y^39+30908255400*y^38+29613703680*y^37+33475986720*y^36+ 32135114880*y^35+34110069840*y^34+32542836480*y^33+34656763680*y^32+31449876480 *y^31+32748575640*y^30+29481408000*y^29+28787057640*y^28+24917365440*y^27+ 23105221920*y^26+18568630080*y^25+15617432160*y^24+11524360320*y^23+8479171800* y^22+5385853572*y^21+3210147600*y^20+1614319740*y^19+697173840*y^18+235077480*y ^17+59457024*y^16+10268280*y^15+1025640*y^14+53460*y^13+1320*y^12+12*y^11] Nevertheless, it is still interesting to ivestigate the statistical properti\ es of the random variable total height, defined on the "sample space" of rooted labeled trees on n ver\ tices. (n - 1) Note that, J[n](1) = n , and the sum of all total heights (that would y\ (n - 1) ield the expectation upon dividing by, n is the FIRST derivative of, J[n](y), with respect to y, evaluated at y=1. This is sequence https://oeis.org/A001864 in the OEIS, whose first 14 terms are: 0, 2, 24, 312, 4720, 82800, 1662024, 37665152, 952401888, 26602156800, 813815035000, 27069937855488, 972940216546896, 37581134047987712 Divided by n (to get the mean height among all vertices) it is https://oeis.\ org/A000435 0, 1, 8, 78, 944, 13800, 237432, 4708144, 105822432, 2660215680, 7398318500\ 0, 2255828154624, 74841555118992, 2684366717713408, 103512489775594200, , that according to the comment there (written by Neil Sloane) it is "The sequ\ ence that started it all" John Riordan and Neil Sloane found an explicit formula for that sequence /n - 2 \ |----- i | | \ n | V[n] = n! | ) ----| | / i! | |----- | \i = 0 / and then used G.N. Watson's elaboration of a formula of Ramanujan to deduce \ that the average total height, that we will call , W[n] /n - 2 \ |----- i | | \ n | n! | ) ----| | / i! | |----- | \i = 0 / W[n] = --------------- (n - 1) n V[n] i.e. , W[n] = --------, is asymptotic to (n - 1) n 1/2 1/2 3/2 2 Pi n --------------- 2 Hence the asymptotic mean height of random rooted labled tree is, 1/2 1/2 1/2 2 Pi n --------------- 2 and in Maple notation 1/2*2^(1/2)*Pi^(1/2)*n^(1/2) The purpose of this article is to state (rigorously proved!) explicit expres\ sions for the variance, and the moments (about the mean) up to the, 12, -th, in terms of n and W[n] as well as their asymptotics (implied by the Watson-Ramanujan formula) and the limits of the standardized moments, that would hopefully lead, one d\ ay, to an explicit evaluation of the probability density function of the scaled-limiting distribution. ----------------------------------------------------------------------------\ ----------------- Theorem 1 (Riordan-Sloane): The expectation of the "total height" in the set\ of rooted labeled trees is W[n] 1/2 1/2 3/2 2 Pi n whose asympotitc expression is, --------------- 2 and in Maple notation 1/2*2^(1/2)*Pi^(1/2)*n^(3/2) All the remaining, 11, theorem are new, as far as I know. In the statements below 1 a = --------- 10 - 3 Pi and in Maple notation a = 1/(10-3*Pi) ----------------------------------------------------------------------------\ ----------------- Theorem 2: The variance of the random variable "total height" on the set of \ rooted labeled trees on n vertices is given explicitly by 3 2 2 5 n 5 n -W[n] + ---- - ---- - 17/6 n W[n] 3 3 and in Maple notation -W[n]^2+5/3*n^3-5/3*n^2-17/6*n*W[n] its asymptotic expression is / Pi \ 3 |- ---- + 5/3| n \ 2 / and in Maple notation (-1/2*Pi+5/3)*n^3 Hence the limit of the coefficient of variation, as n goes to infinity is, 1/2 1/2 (-18 Pi + 60) 2 --------------------- 1/2 6 Pi and in Maple notation 1/6*(-18*Pi+60)^(1/2)*2^(1/2)/Pi^(1/2) that is approximately equal to, 10, significant digits to 0.2470484847 so the limiting coefficient of variation is, 24.70484847, percents, and there is NO concentration about the mean. ----------------------------------------------------------------------------\ ----------------- Theorem 3: The third moment (about the mean) of the random variable "total height" on the set of rooted labeled trees on n verti\ ces is given explicitly by 3 2 3 2 / 3 277 2\ 76 n n 2 W[n] + 17/2 n W[n] + |-25/8 n - 1/60 n + --- n | W[n] + ----- - ---- \ 24 / 15 30 4 151 n - ------ 30 and in Maple notation 2*W[n]^3+17/2*n*W[n]^2+(-25/8*n^3-1/60*n+277/24*n^2)*W[n]+76/15*n^3-1/30*n^2-\ 151/30*n^4 its asymptotic expression is / 1/2 3/2 1/2 1/2\ |2 Pi 25 2 Pi | (9/2) |---------- - -------------| n \ 2 16 / and in Maple notation (1/2*2^(1/2)*Pi^(3/2)-25/16*2^(1/2)*Pi^(1/2))*n^(9/2) in floating point (9/2) 0.020795808 n and in Maple notation .20795808e-1*n^(9/2) Hence the limit of the SKEWNESS, as n goes to infinity is, 1/2 / Pi \1/2 (6 Pi - 75/4) 3 |---------| \10 - 3 Pi/ --------------------------------- 10 - 3 Pi that in Maple notation is (6*Pi-75/4)/(10-3*Pi)*3^(1/2)*(Pi/(10-3*Pi))^(1/2) this is approximately to, 10, digits .7005665208 1/2 (5 a - 8) (10 a - 1) and in terms of a=1/(10-3*Pi) this equals , ----------------------- 4 that in Maple notation is 1/4*(5*a-8)*(10*a-1)^(1/2) hence the limiting distribution is positively skewed, ----------------------------------------------------------------------------\ ----------------- Theorem 4: The fourth moment (about the mean) of the random variable "total height" on the set of rooted labeled trees on n verti\ ces is given explicitly by 4 3 2 3 2 -3 W[n] - 17 W[n] n + (-217/6 n + 1/15 n + 5/2 n ) W[n] 3 2 5 / 74381 3 649 4 433 2 \ 109 n 2 n 4693 n + |- ----- n + --- n + ---- n + 1/105 n| W[n] + ------ + ---- + ------- \ 2160 80 2520 / 1260 105 540 4 6 4651 n 221 n - ------- + ------ 378 63 and in Maple notation -3*W[n]^4-17*W[n]^3*n+(-217/6*n^2+1/15*n+5/2*n^3)*W[n]^2+(-74381/2160*n^3+649/ 80*n^4+433/2520*n^2+1/105*n)*W[n]+109/1260*n^3+2/105*n^2+4693/540*n^5-4651/378* n^4+221/63*n^6 its asymptotic expression is / 2 221\ 6 |-3/4 Pi + 5/4 Pi + ---| n \ 63 / and in Maple notation (-3/4*Pi^2+5/4*Pi+221/63)*n^6 in floating point 6 0.032724023 n and in Maple notation .32724023e-1*n^6 Hence the limit of the KURTOSIS, as n goes to infinity is, 2 -189 Pi + 315 Pi + 884 ----------------------- 2 7 (10 - 3 Pi) that in Maple notation is 1/7*(-189*Pi^2+315*Pi+884)/(10-3*Pi)^2 this is approximately to, 10, digits 3.560394751 2 and in terms of a=1/(10-3*Pi) this equals , -166/7 a + 45 a - 3 that in Maple notation is -166/7*a^2+45*a-3 hence the limiting distribution is LEPTOKURTIC. ----------------------------------------------------------------------------\ ----------------- Theorem Number, 5, The, 5, -th moment (about the mean) of the random variabl\ e "total height" on the set of rooted labeled trees on n vertices is given explicitly by 5 4 /985 2 25 3\ 3 4 W[n] + 85/3 W[n] n + |--- n - 1/6 n + -- n | W[n] \12 12 / / 265 2 52493 3 469 4\ 2 + |- --- n - 1/21 n + ----- n + --- n | W[n] \ 504 432 48 / / 105845 6 27955 5 107 2299 3 6620267 4 161 2\ + |- ------ n + ----- n - ---- n - ---- n + ------- n - ---- n | W[n] \ 8064 3456 9240 5040 72576 1440 / 6 4 5 3 7 2 1617437 n 6431 n 1707337 n 2479 n 228161 n 107 n - ---------- - ------- + ---------- - ------- - --------- - ------ 498960 110880 62370 55440 9504 4620 and in Maple notation 4*W[n]^5+85/3*W[n]^4*n+(985/12*n^2-1/6*n+25/12*n^3)*W[n]^3+(-265/504*n^2-1/21*n +52493/432*n^3+469/48*n^4)*W[n]^2+(-105845/8064*n^6+27955/3456*n^5-107/9240*n-\ 2299/5040*n^3+6620267/72576*n^4-161/1440*n^2)*W[n]-1617437/498960*n^6-6431/ 110880*n^4+1707337/62370*n^5-2479/55440*n^3-228161/9504*n^7-107/4620*n^2 its asymptotic expression is / 1/2 5/2 1/2 3/2 1/2 1/2\ |2 Pi 25 2 Pi 105845 2 Pi | (15/2) |---------- + ------------- - -----------------| n \ 2 48 16128 / and in Maple notation (1/2*2^(1/2)*Pi^(5/2)+25/48*2^(1/2)*Pi^(3/2)-105845/16128*2^(1/2)*Pi^(1/2))*n^( 15/2) in floating point (15/2) 0.02065049 n and in Maple notation .2065049e-1*n^(15/2) Hence the limit of the scaled, 5, -th momemnt , as n goes to infinity is, / 2 105845\ 1/2 / Pi \1/2 |36 Pi + 75/2 Pi - ------| 3 |---------| \ 224 / \10 - 3 Pi/ ----------------------------------------------- 2 (10 - 3 Pi) that in Maple notation is (36*Pi^2+75/2*Pi-105845/224)/(10-3*Pi)^2*3^(1/2)*(Pi/(10-3*Pi))^(1/2) this is approximately to, 10, digits 7.256376376 and in terms of a=1/(10-3*Pi) this equals , 2 1/2 (11755 a - 20720 a + 896) (10 a - 1) ---------------------------------------- 224 that in Maple notation is 1/224*(11755*a^2-20720*a+896)*(10*a-1)^(1/2) ----------------------------------------------------------------------------\ ----------------- Theorem Number, 6, The, 6, -th moment (about the mean) of the random variabl\ e "total height" on the set of rooted labeled trees on n vertices is given explicitly by 6 5 3 2 4 -5 W[n] - 85/2 W[n] n + (-25/2 n - 935/6 n + 1/3 n) W[n] / 45197 3 3823 4 209 2 \ 3 + |- ----- n - ---- n + --- n + 1/7 n| W[n] \ 144 48 168 / /647 2 107 403 3 4387787 4 35125 6 103043 5\ 2 + |---- n + ---- n + --- n - ------- n + ----- n - ------ n | W[n] + \1680 1540 280 12096 1344 576 / / 66616493 5 19843 4 49350407 6 760633 2 16 17048249 7 |- -------- n + ----- n - -------- n + ------- n + --- n + -------- n \ 295680 55440 295680 5045040 693 177408 5 3 7 6 2261099 3\ 16483321 n 117809 n 110011753 n 976717883 n + ------- n | W[n] - ----------- + --------- - ------------ - ------------ 9979200 / 908107200 2522520 2162160 16816800 4 9 2 8 37756501 n 82825 n 32 n 172583989 n - ----------- + -------- + ----- + ------------ 454053600 9009 693 1729728 and in Maple notation -5*W[n]^6-85/2*W[n]^5*n+(-25/2*n^3-935/6*n^2+1/3*n)*W[n]^4+(-45197/144*n^3-3823 /48*n^4+209/168*n^2+1/7*n)*W[n]^3+(647/1680*n^2+107/1540*n+403/280*n^3-4387787/ 12096*n^4+35125/1344*n^6-103043/576*n^5)*W[n]^2+(-66616493/295680*n^5+19843/ 55440*n^4-49350407/295680*n^6+760633/5045040*n^2+16/693*n+17048249/177408*n^7+ 2261099/9979200*n^3)*W[n]-16483321/908107200*n^5+117809/2522520*n^3-110011753/ 2162160*n^7-976717883/16816800*n^6-37756501/454053600*n^4+82825/9009*n^9+32/693 *n^2+172583989/1729728*n^8 its asymptotic expression is /82825 3 2 35125 \ 9 |----- - 5/8 Pi - 25/8 Pi + ----- Pi| n \9009 2688 / and in Maple notation (82825/9009-5/8*Pi^3-25/8*Pi^2+35125/2688*Pi)*n^9 in floating point 9 0.02439526 n and in Maple notation .2439526e-1*n^9 Hence the limit of the scaled, 6, -th momemnt , as n goes to infinity is, 3 2 15 (-144144 Pi - 720720 Pi + 3013725 Pi + 2120320) ---------------------------------------------------- 3 16016 (10 - 3 Pi) that in Maple notation is 15/16016*(-144144*Pi^3-720720*Pi^2+3013725*Pi+2120320)/(10-3*Pi)^3 this is approximately to, 10, digits 27.68549546 and in terms of a=1/(10-3*Pi) this equals , 1264925 3 230625 2 - ------- a + ------ a - 225 a + 5 1144 112 that in Maple notation is -1264925/1144*a^3+230625/112*a^2-225*a+5 ----------------------------------------------------------------------------\ ----------------- Theorem Number, 7, The, 7, -th moment (about the mean) of the random variabl\ e "total height" on the set of rooted labeled trees on n vertices is given explicitly by 7 6 / 3 6335 2\ 5 6 W[n] + 119/2 W[n] n + |-7/12 n + 245/8 n + ---- n | W[n] \ 24 / / 3913 4 290843 3 181 2\ 4 + |-1/3 n + ---- n + ------ n - --- n | W[n] \ 16 432 72 / / 34655 6 2537 3 3643627 4 487 2 2689519 5 107 \ 3 + |- ----- n - ---- n + ------- n - --- n + ------- n - --- n| W[n] \ 1152 720 3456 480 3456 440 / / 20393 4 94006531 6 4271233 7 922439 3 16 410101 2 + |- ----- n + -------- n - ------- n - ------- n - -- n - ------ n \ 15840 76032 25344 1425600 99 720720 381009301 5\ 2 /3305921733221 6 11771797 4 205783 + --------- n | W[n] + |------------- n + -------- n - ------- n 380160 / \ 6227020800 19219200 3063060 6484393 3 2346847 2 33965825 9 371150089 8 19182762787 7 - -------- n - ------- n - -------- n - --------- n + ----------- n 37065600 7207200 658944 988416 19768320 3 8 7 22622767 5\ 48449 n 140927991493 n 1595237098951 n - ---------- n | W[n] - -------- + --------------- + ---------------- 1037836800 / 673200 481178880 13232419200 6 2 4 5 10 400943773 n 205783 n 5168837 n 136752977 n 165677519 n - ------------ - --------- + ---------- + ------------ - ------------- 1260230400 1531530 15422400 735134400 1555840 9 38682150413 n - -------------- 126023040 and in Maple notation 6*W[n]^7+119/2*W[n]^6*n+(-7/12*n+245/8*n^3+6335/24*n^2)*W[n]^5+(-1/3*n+3913/16* n^4+290843/432*n^3-181/72*n^2)*W[n]^4+(-34655/1152*n^6-2537/720*n^3+3643627/ 3456*n^4-487/480*n^2+2689519/3456*n^5-107/440*n)*W[n]^3+(-20393/15840*n^4+ 94006531/76032*n^6-4271233/25344*n^7-922439/1425600*n^3-16/99*n-410101/720720*n ^2+381009301/380160*n^5)*W[n]^2+(3305921733221/6227020800*n^6+11771797/19219200 *n^4-205783/3063060*n-6484393/37065600*n^3-2346847/7207200*n^2-33965825/658944* n^9-371150089/988416*n^8+19182762787/19768320*n^7-22622767/1037836800*n^5)*W[n] -48449/673200*n^3+140927991493/481178880*n^8+1595237098951/13232419200*n^7-\ 400943773/1260230400*n^6-205783/1531530*n^2+5168837/15422400*n^4+136752977/ 735134400*n^5-165677519/1555840*n^10-38682150413/126023040*n^9 its asymptotic expression is / 1/2 7/2 1/2 5/2 1/2 3/2 1/2 1/2\ |3 2 Pi 245 2 Pi 34655 2 Pi 33965825 2 Pi | |------------ + -------------- - ---------------- - -------------------| \ 8 64 4608 1317888 / (21/2) n and in Maple notation (3/8*2^(1/2)*Pi^(7/2)+245/64*2^(1/2)*Pi^(5/2)-34655/4608*2^(1/2)*Pi^(3/2)-\ 33965825/1317888*2^(1/2)*Pi^(1/2))*n^(21/2) in floating point (21/2) 0.02455953 n and in Maple notation .2455953e-1*n^(21/2) Hence the limit of the scaled, 7, -th momemnt , as n goes to infinity is, / 3 2 103965 101897475\ 1/2 / Pi \1/2 |162 Pi + 6615/4 Pi - ------ Pi - ---------| 3 |---------| \ 32 9152 / \10 - 3 Pi/ ------------------------------------------------------------------ 3 (10 - 3 Pi) that in Maple notation is (162*Pi^3+6615/4*Pi^2-103965/32*Pi-101897475/9152)/(10-3*Pi)^3*3^(1/2)*(Pi/(10-\ 3*Pi))^(1/2) this is approximately to, 10, digits 90.01702180 and in terms of a=1/(10-3*Pi) this equals , 3 2 1/2 (22069225 a - 40195870 a + 3329040 a - 54912) (10 a - 1) ------------------------------------------------------------- 9152 that in Maple notation is 1/9152*(22069225*a^3-40195870*a^2+3329040*a-54912)*(10*a-1)^(1/2) ----------------------------------------------------------------------------\ ----------------- Theorem Number, 8, The, 8, -th moment (about the mean) of the random variabl\ e "total height" on the set of rooted labeled trees on n vertices is given explicitly by 8 7 /14 2 3\ 6 -7 W[n] - 238/3 W[n] n + |-- n - 413 n - 175/3 n | W[n] \15 / / 1377607 3 821 2 67151 4\ 5 + |- ------- n + --- n + 2/3 n - ----- n | W[n] \ 1080 180 120 / /332 3 235 6 984109 5 107 407 2 3271547 4\ 4 / + |--- n - --- n - ------ n + --- n + --- n - ------- n | W[n] + | \45 144 432 165 180 1296 / \ 2327 4 476219 3 64 1435976957 6 924488767 5 36683 7 ---- n + ------ n + -- n - ---------- n - --------- n + ----- n 660 356400 99 285120 285120 19008 293257 2\ 3 /2979511 3 33362503 4 25982111 8 + ------ n | W[n] + |-------- n - -------- n + -------- n 180180 / \32432400 12972960 123552 88556051 5 12762625 9 1182047 2 15662386691 7 + --------- n + -------- n + ------- n - ----------- n 129729600 82368 900900 2471040 2040095356853 6 411566 \ 2 / 344993637283 4 95722091 3 - ------------- n + ------ n| W[n] + |- ------------ n - ---------- n 778377600 765765 / \ 132324192000 5145940800 144190248973 10 438745051 2 61467 1597519793 5 + ------------ n + --------- n + ------ n - ---------- n 224040960 436486050 230945 1603929600 16386804121789 8 772540466166743 7 358675227191 9 - -------------- n - --------------- n + ------------ n 3849431040 635156121600 504092160 4 8 117511849889 6\ 68025999889 n 314627356709 n + ------------ n | W[n] - -------------- - --------------- 45368294400 / 48886437600 1269777600 7 6 12 5 44885520498607 n 10762039371793 n 256406305 n 139185587251 n + ----------------- + ----------------- + ------------- - --------------- 105594705216000 8799558768000 8729721 463134672000 3 9 2 10 27646126 n 3534667654890731 n 122934 n 60188263332311 n + ----------- - ------------------- + --------- + ------------------ 218243025 3016991577600 230945 91423987200 11 9802399911719 n + ----------------- 13408851456 and in Maple notation -7*W[n]^8-238/3*W[n]^7*n+(14/15*n-413*n^2-175/3*n^3)*W[n]^6+(-1377607/1080*n^3+ 821/180*n^2+2/3*n-67151/120*n^4)*W[n]^5+(332/45*n^3-235/144*n^6-984109/432*n^5+ 107/165*n+407/180*n^2-3271547/1296*n^4)*W[n]^4+(2327/660*n^4+476219/356400*n^3+ 64/99*n-1435976957/285120*n^6-924488767/285120*n^5+36683/19008*n^7+293257/ 180180*n^2)*W[n]^3+(2979511/32432400*n^3-33362503/12972960*n^4+25982111/123552* n^8+88556051/129729600*n^5+12762625/82368*n^9+1182047/900900*n^2-15662386691/ 2471040*n^7-2040095356853/778377600*n^6+411566/765765*n)*W[n]^2+(-344993637283/ 132324192000*n^4-95722091/5145940800*n^3+144190248973/224040960*n^10+438745051/ 436486050*n^2+61467/230945*n-1597519793/1603929600*n^5-16386804121789/ 3849431040*n^8-772540466166743/635156121600*n^7+358675227191/504092160*n^9+ 117511849889/45368294400*n^6)*W[n]-68025999889/48886437600*n^4-314627356709/ 1269777600*n^8+44885520498607/105594705216000*n^7+10762039371793/8799558768000* n^6+256406305/8729721*n^12-139185587251/463134672000*n^5+27646126/218243025*n^3 -3534667654890731/3016991577600*n^9+122934/230945*n^2+60188263332311/ 91423987200*n^10+9802399911719/13408851456*n^11 its asymptotic expression is /256406305 4 175 3 235 2 12762625 \ 12 |--------- - 7/16 Pi - --- Pi - --- Pi + -------- Pi| n \ 8729721 24 576 164736 / and in Maple notation (256406305/8729721-7/16*Pi^4-175/24*Pi^3-235/576*Pi^2+12762625/164736*Pi)*n^12 in floating point 12 0.0303110 n and in Maple notation .303110e-1*n^12 Hence the limit of the scaled, 8, -th momemnt , as n goes to infinity is, 3 ( 4 3 2 -488864376 Pi - 8147739600 Pi - 455885430 Pi + 86568885375 Pi / 4 + 32820007040) / (2586584 (10 - 3 Pi) ) / that in Maple notation is 3/2586584*(-488864376*Pi^4-8147739600*Pi^3-455885430*Pi^2+86568885375*Pi+ 32820007040)/(10-3*Pi)^4 this is approximately to, 10, digits 358.8086679 and in terms of a=1/(10-3*Pi) this equals , 68706293065 4 115208325 3 2 - ----------- a + --------- a - 59035/4 a + 630 a - 7 1293292 1144 that in Maple notation is -68706293065/1293292*a^4+115208325/1144*a^3-59035/4*a^2+630*a-7 ----------------------------------------------------------------------------\ ----------------- Theorem Number, 9, The, 9, -th moment (about the mean) of the random variabl\ e "total height" on the set of rooted labeled trees on n vertices is given explicitly by 9 8 3 2 7 8 W[n] + 102 W[n] n + (-7/5 n + 195/2 n + 1219/2 n ) W[n] / 153 2 43673 4 265307 3\ 6 + |-6/5 n - --- n + ----- n + ------ n | W[n] \ 20 40 120 / /3048299 4 321 3 7307 6 5183353 5 359 2\ 5 + |------- n - --- n - 111/8 n + ---- n + ------- n - --- n | W[n] + \ 576 220 64 960 80 / / 253109 3 21493 4 2117275 7 64 90579911 5 46967 2 |- ------ n - ----- n + ------- n - -- n + -------- n - ----- n \ 118800 2640 2112 33 10560 12012 160990717 6\ 4 / 2381341 2 1618153231397 6 349411541 4 + --------- n | W[n] + |- ------- n + ------------- n + --------- n 10560 / \ 600600 172972800 43243200 11157569 3 267221645 8 43466783977 7 205783 17084675 9 + -------- n + --------- n + ----------- n - ------ n - -------- n 7207200 82368 1647360 85085 54912 331601437 5\ 3 / 553203 167599151443 4 48759998029 10 - --------- n | W[n] + |- ------ n + ------------ n - ----------- n 86486400 / \ 230945 14702688000 24893440 86746393139 9 466254943168151 7 59775946577 6 4444431931 5 + ----------- n + --------------- n - ----------- n + ---------- n 18670080 70572902400 5040921600 1960358400 11877108394013 8 29164633 2 315419743 3\ 2 / + -------------- n - -------- n + --------- n | W[n] + | 427714560 6928350 114354240 / \ 6978220262525 12 48042559293393293 8 1606145283503303 4 - ------------- n + ----------------- n + ---------------- n 31783944192 17553405542400 140792940288000 57122397595923 10 349259717 8837856747139037 6 49347895529 3 + -------------- n - --------- n - ---------------- n + ----------- n 24078745600 254963280 844757641728000 28117689600 1049060013875906177 9 580650727145131 7 1087789872696497 11 + ------------------- n - --------------- n - ---------------- n 64362486988800 211189410432000 238379581440 4 1274485612091 2 204990691393 5\ 44021189887573 n - ------------- n + ------------ n | W[n] + ----------------- 307286179200 570011904000 / 6184134356400 8 7 6 802627798318267 n 55124685038037037 n 229854815918598671 n + ------------------ - -------------------- - --------------------- 264673292083200 213723683357184000 35620613892864000 10 13 12 564916761805089079 n 278556871026491 n 13464283239070387 n + ---------------------- - ------------------- - --------------------- 138781612569600 548273037312 3524612382720 11 9 5 10083137952101677 n 1533967513080015649 n 4073819431842301 n - --------------------- + ---------------------- - ------------------- 41120477798400 3053195476531200 17810306946432000 3 2 294698706433 n 349259717 n - --------------- - ------------ 3533791060800 127481640 and in Maple notation 8*W[n]^9+102*W[n]^8*n+(-7/5*n+195/2*n^3+1219/2*n^2)*W[n]^7+(-6/5*n-153/20*n^2+ 43673/40*n^4+265307/120*n^3)*W[n]^6+(3048299/576*n^4-321/220*n-111/8*n^3+7307/ 64*n^6+5183353/960*n^5-359/80*n^2)*W[n]^5+(-253109/118800*n^3-21493/2640*n^4+ 2117275/2112*n^7-64/33*n+90579911/10560*n^5-46967/12012*n^2+160990717/10560*n^6 )*W[n]^4+(-2381341/600600*n^2+1618153231397/172972800*n^6+349411541/43243200*n^ 4+11157569/7207200*n^3+267221645/82368*n^8+43466783977/1647360*n^7-205783/85085 *n-17084675/54912*n^9-331601437/86486400*n^5)*W[n]^3+(-553203/230945*n+ 167599151443/14702688000*n^4-48759998029/24893440*n^10+86746393139/18670080*n^9 +466254943168151/70572902400*n^7-59775946577/5040921600*n^6+4444431931/ 1960358400*n^5+11877108394013/427714560*n^8-29164633/6928350*n^2+315419743/ 114354240*n^3)*W[n]^2+(-6978220262525/31783944192*n^12+48042559293393293/ 17553405542400*n^8+1606145283503303/140792940288000*n^4+57122397595923/ 24078745600*n^10-349259717/254963280*n-8837856747139037/844757641728000*n^6+ 49347895529/28117689600*n^3+1049060013875906177/64362486988800*n^9-\ 580650727145131/211189410432000*n^7-1087789872696497/238379581440*n^11-\ 1274485612091/307286179200*n^2+204990691393/570011904000*n^5)*W[n]+ 44021189887573/6184134356400*n^4+802627798318267/264673292083200*n^8-\ 55124685038037037/213723683357184000*n^7-229854815918598671/35620613892864000*n ^6+564916761805089079/138781612569600*n^10-278556871026491/548273037312*n^13-\ 13464283239070387/3524612382720*n^12-10083137952101677/41120477798400*n^11+ 1533967513080015649/3053195476531200*n^9-4073819431842301/17810306946432000*n^5 -294698706433/3533791060800*n^3-349259717/127481640*n^2 its asymptotic expression is / 1/2 9/2 1/2 7/2 1/2 5/2 1/2 3/2 |2 Pi 195 2 Pi 7307 2 Pi 17084675 2 Pi |---------- + -------------- + --------------- - ------------------- \ 4 32 512 219648 1/2 1/2\ 6978220262525 2 Pi | (27/2) - ------------------------| n 63567888384 / and in Maple notation (1/4*2^(1/2)*Pi^(9/2)+195/32*2^(1/2)*Pi^(7/2)+7307/512*2^(1/2)*Pi^(5/2)-\ 17084675/219648*2^(1/2)*Pi^(3/2)-6978220262525/63567888384*2^(1/2)*Pi^(1/2))*n^ (27/2) in floating point (27/2) 0.0382071 n and in Maple notation .382071e-1*n^(27/2) Hence the limit of the scaled, 9, -th momemnt , as n goes to infinity is, / 4 3 591867 2 461286225 188411947088175\ 1/2 |648 Pi + 15795 Pi + ------ Pi - --------- Pi - ---------------| 3 \ 16 2288 662165504 / / Pi \1/2 / 4 |---------| / (10 - 3 Pi) \10 - 3 Pi/ / that in Maple notation is (648*Pi^4+15795*Pi^3+591867/16*Pi^2-461286225/2288*Pi-188411947088175/662165504 )/(10-3*Pi)^4*3^(1/2)*(Pi/(10-3*Pi))^(1/2) this is approximately to, 10, digits 1460.710269 4 and in terms of a=1/(10-3*Pi) this equals , (79090804803025 a 3 2 - 147331855027840 a + 17521023391872 a - 599259781120 a + 5297324032) 1/2 (10 a - 1) /662165504 that in Maple notation is 1/662165504*(79090804803025*a^4-147331855027840*a^3+17521023391872*a^2-\ 599259781120*a+5297324032)*(10*a-1)^(1/2) ----------------------------------------------------------------------------\ ----------------- Theorem Number, 10, The, 10, -th moment (about the mean) of the random varia\ ble "total height" on the set of rooted labeled trees on n vertices is given explicitly by 10 9 3 2 8 -9 W[n] - 255/2 W[n] n + (-150 n - 860 n + 2 n) W[n] / 258011 3 4 145 2\ 7 + |2 n - ------ n - 15363/8 n + --- n | W[n] \ 72 12 / / 36065 6 327 2 3215135 5 321 322163 4 1447 3\ 6 + |- ----- n + --- n - ------- n + --- n - ------ n + ---- n | W[n] \ 96 40 288 110 32 60 / / 2427800251 6 119243 3 1249429097 5 16974883 7 160 + |- ---------- n + ------ n - ---------- n - -------- n + --- n \ 63360 47520 63360 4224 33 22043 4 998909 2\ 5 / 366504695 8 1656919 3 + ----- n + ------ n | W[n] + |- --------- n - ------- n 1320 120120 / \ 20592 196560 13902198643 7 30433171 4 10805125 9 411566 121522693 5 - ----------- n - -------- n + -------- n + ------ n + --------- n 164736 1441440 27456 51051 8648640 599647 2 1407182168669 6\ 4 / 364159768835287 7 + ------ n - ------------- n | W[n] + |- --------------- n 60060 51891840 / \ 14114580480 16949914381 10 1399631944859 9 31121629454263 8 40530645473 6 + ----------- n - ------------- n - -------------- n + ----------- n 4978688 33606144 256628736 1008184320 1084709 5 2070890923 3 125954891 2 553203 108467656163 4\ - ------- n - ---------- n + --------- n + ------ n - ------------ n | 6223360 114354240 9699690 46189 2940537600 / 3 /303597879758977 11 4188655738524461 6 13886296807025 12 W[n] + |--------------- n + ---------------- n + -------------- n \ 23837958144 84475764172800 15891972096 44170260664417 7 55732948072451 4 709731919006396001 9 + -------------- n - -------------- n - ------------------ n 5279735260800 1083022617600 6436248698880 4067174383891403 8 1111394441849 3 349259717 538415828411 2 - ---------------- n - ------------- n + --------- n + ------------ n 250762936320 47800072320 25496328 30728617920 7356483293963 5 3468329161723873 10\ 2 /131729093 + ------------- n - ---------------- n | W[n] + |--------- n 741015475200 65012613120 / \14872858 1967276319524153 12 6823145862314711687 10 6704616275984551333 11 + ---------------- n - ------------------- n - ------------------- n 85445148672 118426976059392 197378293432320 37227227469313517617 6 17834911704985315 13 541491031583180687 4 + -------------------- n + ----------------- n - ------------------ n 641171050071552000 4386184298496 8905153473216000 4593672500051 2 1199208956605249 7 2338922889860231579 5 + ------------- n - ---------------- n + ------------------- n 210179729760 379953214857216 152659773826560000 4961900632643393 3 11319909826001593 8 15935377833611136665887 9\ - ---------------- n - ----------------- n - ----------------------- n | 296838449107200 397009938124800 2637960891722956800 / 7 15 12 287700013383470112443 n 304702375 n 29685100324638471751 n W[n] - ------------------------ + ------------- - ------------------------ 41676118254650880000 2725569 3996910442004480 8 10 1558454676877934757707 n 5056483411604494577 n - ------------------------- - ----------------------- 83352236509301760000 5020043019264000 13 14 36359326872301497943 n 3770746846608494867 n + ------------------------ + ----------------------- 2220505801113600 740168600371200 11 5 1969702150214007503933 n 279760026310461467 n - -------------------------- + --------------------- 149884141575168000 28941748787952000 9 6 2 1175474728490347469 n 39162636389390585447 n 131729093 n + ---------------------- + ----------------------- + ------------ 952596988677734400 992288529872640000 7436429 3 4 1324927442911 n 86958363099381089 n - ---------------- - -------------------- 595509234320 1929449919196800 and in Maple notation -9*W[n]^10-255/2*W[n]^9*n+(-150*n^3-860*n^2+2*n)*W[n]^8+(2*n-258011/72*n^3-\ 15363/8*n^4+145/12*n^2)*W[n]^7+(-36065/96*n^6+327/40*n^2-3215135/288*n^5+321/ 110*n-322163/32*n^4+1447/60*n^3)*W[n]^6+(-2427800251/63360*n^6+119243/47520*n^3 -1249429097/63360*n^5-16974883/4224*n^7+160/33*n+22043/1320*n^4+998909/120120*n ^2)*W[n]^5+(-366504695/20592*n^8-1656919/196560*n^3-13902198643/164736*n^7-\ 30433171/1441440*n^4+10805125/27456*n^9+411566/51051*n+121522693/8648640*n^5+ 599647/60060*n^2-1407182168669/51891840*n^6)*W[n]^4+(-364159768835287/ 14114580480*n^7+16949914381/4978688*n^10-1399631944859/33606144*n^9-\ 31121629454263/256628736*n^8+40530645473/1008184320*n^6-1084709/6223360*n^5-\ 2070890923/114354240*n^3+125954891/9699690*n^2+553203/46189*n-108467656163/ 2940537600*n^4)*W[n]^3+(303597879758977/23837958144*n^11+4188655738524461/ 84475764172800*n^6+13886296807025/15891972096*n^12+44170260664417/5279735260800 *n^7-55732948072451/1083022617600*n^4-709731919006396001/6436248698880*n^9-\ 4067174383891403/250762936320*n^8-1111394441849/47800072320*n^3+349259717/ 25496328*n+538415828411/30728617920*n^2+7356483293963/741015475200*n^5-\ 3468329161723873/65012613120*n^10)*W[n]^2+(131729093/14872858*n+ 1967276319524153/85445148672*n^12-6823145862314711687/118426976059392*n^10-\ 6704616275984551333/197378293432320*n^11+37227227469313517617/ 641171050071552000*n^6+17834911704985315/4386184298496*n^13-541491031583180687/ 8905153473216000*n^4+4593672500051/210179729760*n^2-1199208956605249/ 379953214857216*n^7+2338922889860231579/152659773826560000*n^5-4961900632643393 /296838449107200*n^3-11319909826001593/397009938124800*n^8-\ 15935377833611136665887/2637960891722956800*n^9)*W[n]-287700013383470112443/ 41676118254650880000*n^7+304702375/2725569*n^15-29685100324638471751/ 3996910442004480*n^12-1558454676877934757707/83352236509301760000*n^8-\ 5056483411604494577/5020043019264000*n^10+36359326872301497943/2220505801113600 *n^13+3770746846608494867/740168600371200*n^14-1969702150214007503933/ 149884141575168000*n^11+279760026310461467/28941748787952000*n^5+ 1175474728490347469/952596988677734400*n^9+39162636389390585447/ 992288529872640000*n^6+131729093/7436429*n^2-1324927442911/595509234320*n^3-\ 86958363099381089/1929449919196800*n^4 its asymptotic expression is /304702375 5 36065 3 10805125 2 13886296807025 4 |--------- - 9/32 Pi - ----- Pi + -------- Pi + -------------- Pi - 75/8 Pi \ 2725569 768 109824 31783944192 \ 15 | n / and in Maple notation (304702375/2725569-9/32*Pi^5-36065/768*Pi^3+10805125/109824*Pi^2+13886296807025 /31783944192*Pi-75/8*Pi^4)*n^15 in floating point 15 0.0526277 n and in Maple notation .526277e-1*n^15 Hence the limit of the scaled, 10, -th momemnt , as n goes to infinity is, 3 ( 5 4 3 -5551264502784 Pi - 185042150092800 Pi - 926881269874560 Pi 2 / + 1941922651536000 Pi + 8623390317162525 Pi + 2206571720704000) / ( / 5 7614903296 (10 - 3 Pi) ) that in Maple notation is 3/7614903296*(-5551264502784*Pi^5-185042150092800*Pi^4-926881269874560*Pi^3+ 1941922651536000*Pi^2+8623390317162525*Pi+2206571720704000)/(10-3*Pi)^5 this is approximately to, 10, digits 6498.233818 10394988504651375 5 and in terms of a=1/(10-3*Pi) this equals , - ----------------- a 3807451648 1746385189154325 4 1087630425 3 2 + ---------------- a - ---------- a + 468195/8 a - 1350 a + 9 331082752 1144 that in Maple notation is -10394988504651375/3807451648*a^5+1746385189154325/331082752*a^4-1087630425/ 1144*a^3+468195/8*a^2-1350*a+9 ----------------------------------------------------------------------------\ ----------------- Theorem Number, 11, The, 11, -th moment (about the mean) of the random varia\ ble "total height" on the set of rooted labeled trees on n vertices is given explicitly by 10 /5225 3 28105 2 \ 9 935/6 W[n] n + |---- n + ----- n - 11/4 n| W[n] \ 24 24 / / 396847 3 1529 2 4\ 8 + |-22/7 n + ------ n - ---- n + 25091/8 n | W[n] + \ 72 84 / /12064151 5 215073199 4 1184975 6 33143 3 107 23419 2\ |-------- n + --------- n + ------- n - ----- n - --- n - ----- n | \ 576 12096 1344 840 20 1680 / 7 / 1461855949 6 176413 2 7531 4 12740333 7 W[n] + |-32/3 n + ---------- n - ------ n - ---- n + -------- n \ 17280 10920 240 1152 705949631 5 29999 3\ 6 / 205783 203525 9 905593379 8 + --------- n - ----- n | W[n] + |- ------ n - ------ n + --------- n 17280 21600 / \ 9282 9984 14976 67750805167 7 6402997655161 6 640580107 5 22467233 3 + ----------- n + ------------- n - --------- n + -------- n 299520 94348800 15724800 786240 380984563 4 483167 2\ 5 / 423942193 5 2317787027 3 + --------- n - ------ n | W[n] + |- --------- n + ---------- n 7862400 21840 / \ 21385728 31187520 30907994921 6 78901908523 4 86856121 2 62622436333771 7 - ----------- n + ----------- n - -------- n + -------------- n 274959360 801964800 2645370 769886208 105844865831375 8 184401 1044872557 10 569696382721 9\ 4 + --------------- n - ------ n - ---------- n + ------------ n | W[n] 256628736 4199 1357824 3055104 / / 126601646339411 11 596622554049892609 9 42024481161847 7 + |- --------------- n + ------------------ n - -------------- n \ 13002522624 1170227036160 3839807462400 2638922068986269 6 20654085915475 12 61522286488769 4 - ---------------- n - -------------- n + -------------- n 15359229849600 8668348416 365695948800 349259717 241407186003741967 8 3606663543403 3 - --------- n + ------------------ n + ------------- n 4635696 3510681108480 26072766720 135641131029861281 10 10696937387987 5 293059233851 2\ 3 / + ------------------ n - -------------- n - ------------ n | W[n] + | 390075678720 134730086400 5587021440 / \ 514818220301975 13 11062985588015501 12 6946611586834991581 11 - --------------- n - ----------------- n + ------------------- n 30672617472 256335446016 17943481221120 9308638177105469062207 9 131729093 16540294036639637227 6 + ---------------------- n - --------- n - -------------------- n 239814626520268800 1352078 58288277279232000 15150314399468930753 5 380071385312255099 7 29147175760487 2 - -------------------- n + ------------------ n - -------------- n 97147128798720000 7771770303897600 324823218720 224538464392655087 4 66192515179194974249 10 10600402677229181 8 + ------------------ n + -------------------- n + ----------------- n 809559406656000 161491330990080 72183625113600 5085674089345253 3\ 2 / 45997602831105681797 14 + ---------------- n | W[n] + |- -------------------- n 26985313555200 / \ 1076608873267200 2245471489530565679 4 882523729466125 15 4160244721778753 2 + ------------------- n - --------------- n - ---------------- n 5666915846592000 844399116288 29234089684800 907486003367 1694326814934627920663 12 - ------------ n + ---------------------- n 12939386460 7267109894553600 84364007757560022967981 11 180808758591341897 3 + ----------------------- n + ------------------ n 436026593673216000 1169363587392000 1863826568040605243513 6 618255025602276947567 8 - ---------------------- n + --------------------- n 5051650697533440000 3367767131688960000 1246078372890135877289 7 264046404280220011003 13 + ---------------------- n - --------------------- n 11366214069450240000 3229826619801600 3396688679511970185157 5 1690465710960444628187 9 - ---------------------- n - ---------------------- n 17680777441367040000 72743770044481536000 6325735669138592254007 10\ 11 + ---------------------- n | W[n] + 10 W[n] 481924129849344000 / 13 12 1006192095277276867163 n 325735149098523520422197 n + -------------------------- + ---------------------------- 17917692007219200 8024566348947456000 11 10 334220303165541969002863 n 40578958552223965441171 n + ---------------------------- - --------------------------- 163673482600083456000 1054784665644982272000 9 14 3829152863878599393083 n 7065329667834725441617 n - ------------------------- - -------------------------- 104879157095381760000 122935275716198400 15 16 3324889500766867051 n 4190587459669651501 n - ----------------------- - ----------------------- 86089128652800 1561082866237440 8 7 119378807620381363983479 n 6515454059627269568863 n + --------------------------- + ------------------------- 922936582439359488000 64092818224955520000 6 4 73575271256917195560173 n 536278827095095843 n - -------------------------- + --------------------- 256371272899822080000 1541433819744000 5 3 2 53146112288848027367 n 14818899516223643 n 907486003367 n - ----------------------- + -------------------- - --------------- 427285454833036800 423894300429600 6469693230 and in Maple notation 935/6*W[n]^10*n+(5225/24*n^3+28105/24*n^2-11/4*n)*W[n]^9+(-22/7*n+396847/72*n^3 -1529/84*n^2+25091/8*n^4)*W[n]^8+(12064151/576*n^5+215073199/12096*n^4+1184975/ 1344*n^6-33143/840*n^3-107/20*n-23419/1680*n^2)*W[n]^7+(-32/3*n+1461855949/ 17280*n^6-176413/10920*n^2-7531/240*n^4+12740333/1152*n^7+705949631/17280*n^5-\ 29999/21600*n^3)*W[n]^6+(-205783/9282*n-203525/9984*n^9+905593379/14976*n^8+ 67750805167/299520*n^7+6402997655161/94348800*n^6-640580107/15724800*n^5+ 22467233/786240*n^3+380984563/7862400*n^4-483167/21840*n^2)*W[n]^5+(-423942193/ 21385728*n^5+2317787027/31187520*n^3-30907994921/274959360*n^6+78901908523/ 801964800*n^4-86856121/2645370*n^2+62622436333771/769886208*n^7+105844865831375 /256628736*n^8-184401/4199*n-1044872557/1357824*n^10+569696382721/3055104*n^9)* W[n]^4+(-126601646339411/13002522624*n^11+596622554049892609/1170227036160*n^9-\ 42024481161847/3839807462400*n^7-2638922068986269/15359229849600*n^6-\ 20654085915475/8668348416*n^12+61522286488769/365695948800*n^4-349259717/ 4635696*n+241407186003741967/3510681108480*n^8+3606663543403/26072766720*n^3+ 135641131029861281/390075678720*n^10-10696937387987/134730086400*n^5-\ 293059233851/5587021440*n^2)*W[n]^3+(-514818220301975/30672617472*n^13-\ 11062985588015501/256335446016*n^12+6946611586834991581/17943481221120*n^11+ 9308638177105469062207/239814626520268800*n^9-131729093/1352078*n-\ 16540294036639637227/58288277279232000*n^6-15150314399468930753/ 97147128798720000*n^5+380071385312255099/7771770303897600*n^7-29147175760487/ 324823218720*n^2+224538464392655087/809559406656000*n^4+66192515179194974249/ 161491330990080*n^10+10600402677229181/72183625113600*n^8+5085674089345253/ 26985313555200*n^3)*W[n]^2+(-45997602831105681797/1076608873267200*n^14+ 2245471489530565679/5666915846592000*n^4-882523729466125/844399116288*n^15-\ 4160244721778753/29234089684800*n^2-907486003367/12939386460*n+ 1694326814934627920663/7267109894553600*n^12+84364007757560022967981/ 436026593673216000*n^11+180808758591341897/1169363587392000*n^3-\ 1863826568040605243513/5051650697533440000*n^6+618255025602276947567/ 3367767131688960000*n^8+1246078372890135877289/11366214069450240000*n^7-\ 264046404280220011003/3229826619801600*n^13-3396688679511970185157/ 17680777441367040000*n^5-1690465710960444628187/72743770044481536000*n^9+ 6325735669138592254007/481924129849344000*n^10)*W[n]+10*W[n]^11+ 1006192095277276867163/17917692007219200*n^13+325735149098523520422197/ 8024566348947456000*n^12+334220303165541969002863/163673482600083456000*n^11-\ 40578958552223965441171/1054784665644982272000*n^10-3829152863878599393083/ 104879157095381760000*n^9-7065329667834725441617/122935275716198400*n^14-\ 3324889500766867051/86089128652800*n^15-4190587459669651501/1561082866237440*n^ 16+119378807620381363983479/922936582439359488000*n^8+6515454059627269568863/ 64092818224955520000*n^7-73575271256917195560173/256371272899822080000*n^6+ 536278827095095843/1541433819744000*n^4-53146112288848027367/427285454833036800 *n^5+14818899516223643/423894300429600*n^3-907486003367/6469693230*n^2 its asymptotic expression is / 1/2 3/2 1/2 1/2 1/2 11/2 | 20654085915475 2 Pi 882523729466125 2 Pi 5 2 Pi |- ------------------------- - -------------------------- + ------------- \ 34673393664 1688798232576 32 1/2 9/2 1/2 7/2 1/2 5/2\ 5225 2 Pi 1184975 2 Pi 203525 2 Pi | (33/2) + --------------- + ------------------ - -----------------| n 768 21504 79872 / and in Maple notation (-20654085915475/34673393664*2^(1/2)*Pi^(3/2)-882523729466125/1688798232576*2^( 1/2)*Pi^(1/2)+5/32*2^(1/2)*Pi^(11/2)+5225/768*2^(1/2)*Pi^(9/2)+1184975/21504*2^ (1/2)*Pi^(7/2)-203525/79872*2^(1/2)*Pi^(5/2))*n^(33/2) in floating point (33/2) 0.07620704 n and in Maple notation .7620704e-1*n^(33/2) / Hence the limit of the scaled, 11, -th momemnt , as n goes to infinity is, | \ 5 4 95982975 3 16485525 2 557660319717825 2430 Pi + 423225/4 Pi + -------- Pi - -------- Pi - --------------- Pi 112 416 60196864 2647571188398375\ 1/2 / Pi \1/2 / 5 - ----------------| 3 |---------| / (10 - 3 Pi) 325771264 / \10 - 3 Pi/ / that in Maple notation is (2430*Pi^5+423225/4*Pi^4+95982975/112*Pi^3-16485525/416*Pi^2-557660319717825/ 60196864*Pi-2647571188398375/325771264)/(10-3*Pi)^5*3^(1/2)*(Pi/(10-3*Pi))^(1/2 ) this is approximately to, 10, digits 30389.98955 5 and in terms of a=1/(10-3*Pi) this equals , 5 (7039700396762925 a 4 3 2 - 13370192732090140 a + 2028675442090240 a - 104105864778240 a 1/2 + 2000642775040 a - 11076222976) (10 a - 1) /5538111488 that in Maple notation is 5/5538111488*(7039700396762925*a^5-13370192732090140*a^4+2028675442090240*a^3-\ 104105864778240*a^2+2000642775040*a-11076222976)*(10*a-1)^(1/2) ----------------------------------------------------------------------------\ ----------------- Theorem Number, 12, The, 12, -th moment (about the mean) of the random varia\ ble "total height" on the set of rooted labeled trees on n vertices is given explicitly by 12 2 3 10 -11 W[n] + (-9295/6 n + 11/3 n - 605/2 n ) W[n] /4433 2 130053 3 232463 4\ 9 + |---- n + 33/7 n - ------ n - ------ n | W[n] + \168 16 48 / / 197065 6 29847697 4 451 2 1752949 5 8613 3 321 \ 8 |- ------ n - -------- n + --- n - ------- n + ---- n + --- n| W[n] \ 112 1008 20 48 140 35 / / 236227 3 4855207 7 227974903 6 524789809 5 + |- ------ n + 64/3 n - ------- n - --------- n - --------- n \ 75600 192 1344 6720 23143 4 159721 2\ 7 / 39946407881 7 1196211105941 6 + ----- n + ------ n | W[n] + |- ----------- n - ------------- n 420 5460 / \ 74880 7862400 4118525 9 411566 612389623 8 1216541 2 25294649 3 - ------- n + ------ n - --------- n + ------- n - -------- n 2496 7735 3744 27300 327600 397534721 5 28340791 4\ 6 /125672022949 6 553203 + --------- n - -------- n | W[n] + |------------ n + ------ n 3931200 280800 / \ 458265600 4199 42490762687 10 350882124101 9 1320207517 5 72060143236301 8 - ----------- n - ------------ n + ---------- n - -------------- n 2263040 565760 13708800 61102080 1412418146844979 7 8737494277 4 3729227 2 12329623447 3\ 5 - ---------------- n - ---------- n + ------- n - ----------- n | W[n] 6415718400 38188800 51870 51979200 / / 88498116709783 11 3383894554225 12 349259717 + |- -------------- n + ------------- n + --------- n \ 1083543552 722362368 1158924 1247634550777 3 16248251708483113 10 283719845661143 4 - ------------- n - ----------------- n - --------------- n 2172730560 10835435520 639967910400 170380936571 2 41104913045158787 8 50607599172127 7 + ------------ n - ----------------- n - -------------- n 1396755360 175534055424 1919903731200 540067871571640913 9 12367164434999 5 1864055234217173 6\ 4 - ------------------ n + -------------- n + ---------------- n | W[n] 292556759040 33682521600 3839807462400 / /395187279 20668176192235045 12 9644649559081677097 6 + |--------- n - ----------------- n + ------------------- n \ 676039 128167723008 9714712879872000 334632223354847857 11 162422872782059221691 10 - ------------------ n - --------------------- n 142408581120 80745665495040 118887608662479887 4 38496270541081 2 8935635750717385 13 - ------------------ n + -------------- n + ---------------- n 134926567776000 162411609360 199372013568 1250463526012839371 7 7099724958270246527647 9 40662121516283 8 - ------------------- n - ---------------------- n - -------------- n 3885885151948800 39969104420044800 74262988800 14742932456284700653 5 466084718931443 3\ 3 / + -------------------- n - --------------- n | W[n] + | 16191188133120000 408868387200 / \ 52848773354135902905053 11 1397475811688243203153 12 - ----------------------- n - ---------------------- n 36335549472768000 605592491212800 166499542207903097 3 907486003367 26828210698191972541 13 - ------------------ n + ------------ n - -------------------- n 97446965616000 1078282205 269152218316800 2447385876664623012689 8 3655912427811419117351 10 - ---------------------- n - ---------------------- n 2525825348766720000 40160344154112000 363330383418125 15 1217817542941363297 14 + --------------- n + ------------------- n 70366593024 6901338931200 2456694991108819656037 5 1629056705629861417249 7 + ---------------------- n - ---------------------- n 1473398120113920000 1894369011575040000 1196766324994498691207 9 1312050964569953 2 + ---------------------- n + ---------------- n 6061980837040128000 2436174140400 255757583045889616999 6 475211874479448919 4\ 2 / + --------------------- n - ------------------ n | W[n] + | 140323630487040000 266919949296000 / \ 2215606137591034297992529 10 13422574094796522337285969 12 ------------------------- n - -------------------------- n 5424606851888480256000 21398843597193216000 257794855626421583077 3 397705606965503949702371 13 - --------------------- n - ------------------------ n 163199305665396000 310573328125132800 44096345084769466458900943 7 39856433907058333628461 4 - -------------------------- n - ----------------------- n 30764552747978649600000 12818563644991104000 4948648533484584497030831 9 447683089466782718235689 8 + ------------------------- n - ------------------------ n 10068399081156648960000 341828363866429440000 184839737444355677 2 111024594157417236077 16 + ------------------ n + --------------------- n 165639369495600 4162887643299840 1213457989788646016897 15 146650898693815327050521 14 + ---------------------- n + ------------------------ n 3673136155852800 1311309607639449600 1381815543468033192586967 5 18232621555171 + ------------------------- n + -------------- n 640928182249555200000 27349157745 74909674651232800993894471 11 126866942697079072643096621 6\ - -------------------------- n + --------------------------- n | W[n] 2618775721601335296000 46146829121967974400000 / 13 11 111654668006013210234880879 n - 187 W[n] n - ------------------------------- 916489946169262080000 12 11 55419449559319901391894613 n 6101046337682974405791409 n - ------------------------------ - ----------------------------- 13376587350679547904000 53506349402718191616000 10 9 18118028384439241560909491 n 525448687822455431205520949 n + ------------------------------ + ------------------------------ 63112575122691494400000 1049071248706071951360000 14 15 4534991454446322995894170547 n 1355550356081546306856497 n - -------------------------------- + ----------------------------- 15091534446920515584000 9315762004271923200 16 17 152031577863417206585041 n 2147803580425113923599 n + ---------------------------- + -------------------------- 623823348500352000 58497720591974400 18 8 89585870372335 n 401507294281744246947645839 n + ------------------ - ------------------------------ 180504441117 393401718264776981760000 7 6 9990775075380063197708515727 n 4365576906349078479828931 n - ------------------------------- + ---------------------------- 7868034365295539635200000 1762552501186276800000 4 5 12221217457564000670413 n 3349520086450387103842111 n - -------------------------- + ---------------------------- 3794383856720457000 2185565101470983232000 3 2 658799603051187371 n 36465243110342 n - --------------------- + ----------------- 1407934640712600 27349157745 and in Maple notation -11*W[n]^12+(-9295/6*n^2+11/3*n-605/2*n^3)*W[n]^10+(4433/168*n^2+33/7*n-130053/ 16*n^3-232463/48*n^4)*W[n]^9+(-197065/112*n^6-29847697/1008*n^4+451/20*n^2-\ 1752949/48*n^5+8613/140*n^3+321/35*n)*W[n]^8+(-236227/75600*n^3+64/3*n-4855207/ 192*n^7-227974903/1344*n^6-524789809/6720*n^5+23143/420*n^4+159721/5460*n^2)*W[ n]^7+(-39946407881/74880*n^7-1196211105941/7862400*n^6-4118525/2496*n^9+411566/ 7735*n-612389623/3744*n^8+1216541/27300*n^2-25294649/327600*n^3+397534721/ 3931200*n^5-28340791/280800*n^4)*W[n]^6+(125672022949/458265600*n^6+553203/4199 *n-42490762687/2263040*n^10-350882124101/565760*n^9+1320207517/13708800*n^5-\ 72060143236301/61102080*n^8-1412418146844979/6415718400*n^7-8737494277/38188800 *n^4+3729227/51870*n^2-12329623447/51979200*n^3)*W[n]^5+(-88498116709783/ 1083543552*n^11+3383894554225/722362368*n^12+349259717/1158924*n-1247634550777/ 2172730560*n^3-16248251708483113/10835435520*n^10-283719845661143/639967910400* n^4+170380936571/1396755360*n^2-41104913045158787/175534055424*n^8-\ 50607599172127/1919903731200*n^7-540067871571640913/292556759040*n^9+ 12367164434999/33682521600*n^5+1864055234217173/3839807462400*n^6)*W[n]^4+( 395187279/676039*n-20668176192235045/128167723008*n^12+9644649559081677097/ 9714712879872000*n^6-334632223354847857/142408581120*n^11-162422872782059221691 /80745665495040*n^10-118887608662479887/134926567776000*n^4+38496270541081/ 162411609360*n^2+8935635750717385/199372013568*n^13-1250463526012839371/ 3885885151948800*n^7-7099724958270246527647/39969104420044800*n^9-\ 40662121516283/74262988800*n^8+14742932456284700653/16191188133120000*n^5-\ 466084718931443/408868387200*n^3)*W[n]^3+(-52848773354135902905053/ 36335549472768000*n^11-1397475811688243203153/605592491212800*n^12-\ 166499542207903097/97446965616000*n^3+907486003367/1078282205*n-\ 26828210698191972541/269152218316800*n^13-2447385876664623012689/ 2525825348766720000*n^8-3655912427811419117351/40160344154112000*n^10+ 363330383418125/70366593024*n^15+1217817542941363297/6901338931200*n^14+ 2456694991108819656037/1473398120113920000*n^5-1629056705629861417249/ 1894369011575040000*n^7+1196766324994498691207/6061980837040128000*n^9+ 1312050964569953/2436174140400*n^2+255757583045889616999/140323630487040000*n^6 -475211874479448919/266919949296000*n^4)*W[n]^2+(2215606137591034297992529/ 5424606851888480256000*n^10-13422574094796522337285969/21398843597193216000*n^ 12-257794855626421583077/163199305665396000*n^3-397705606965503949702371/ 310573328125132800*n^13-44096345084769466458900943/30764552747978649600000*n^7-\ 39856433907058333628461/12818563644991104000*n^4+4948648533484584497030831/ 10068399081156648960000*n^9-447683089466782718235689/341828363866429440000*n^8+ 184839737444355677/165639369495600*n^2+111024594157417236077/4162887643299840*n ^16+1213457989788646016897/3673136155852800*n^15+146650898693815327050521/ 1311309607639449600*n^14+1381815543468033192586967/640928182249555200000*n^5+ 18232621555171/27349157745*n-74909674651232800993894471/2618775721601335296000* n^11+126866942697079072643096621/46146829121967974400000*n^6)*W[n]-187*W[n]^11* n-111654668006013210234880879/916489946169262080000*n^13-\ 55419449559319901391894613/13376587350679547904000*n^12-\ 6101046337682974405791409/53506349402718191616000*n^11+ 18118028384439241560909491/63112575122691494400000*n^10+ 525448687822455431205520949/1049071248706071951360000*n^9-\ 4534991454446322995894170547/15091534446920515584000*n^14+ 1355550356081546306856497/9315762004271923200*n^15+152031577863417206585041/ 623823348500352000*n^16+2147803580425113923599/58497720591974400*n^17+ 89585870372335/180504441117*n^18-401507294281744246947645839/ 393401718264776981760000*n^8-9990775075380063197708515727/ 7868034365295539635200000*n^7+4365576906349078479828931/1762552501186276800000* n^6-12221217457564000670413/3794383856720457000*n^4+3349520086450387103842111/ 2185565101470983232000*n^5-658799603051187371/1407934640712600*n^3+ 36465243110342/27349157745*n^2 its asymptotic expression is / 11 6 605 5 197065 4 4118525 3 3383894554225 2 |- -- Pi - --- Pi - ------ Pi - ------- Pi + ------------- Pi \ 64 64 1792 19968 2889449472 363330383418125 89585870372335\ 18 + --------------- Pi + --------------| n 140733186048 180504441117 / and in Maple notation (-11/64*Pi^6-605/64*Pi^5-197065/1792*Pi^4-4118525/19968*Pi^3+3383894554225/ 2889449472*Pi^2+363330383418125/140733186048*Pi+89585870372335/180504441117)*n^ 18 in floating point 18 0.1168730 n and in Maple notation .1168730*n^18 Hence the limit of the scaled, 12, -th momemnt , as n goes to infinity is, 27 ( 6 5 4 -4066404049483776 Pi - 223652222721607680 Pi - 2601772448089351680 Pi 3 2 - 4879832965588627200 Pi + 27707639928293288700 Pi / + 61080560747571248125 Pi + 11742199201442693120) / (13691596126208 / 6 (10 - 3 Pi) ) that in Maple notation is 27/13691596126208*(-4066404049483776*Pi^6-223652222721607680*Pi^5-\ 2601772448089351680*Pi^4-4879832965588627200*Pi^3+27707639928293288700*Pi^2+ 61080560747571248125*Pi+11742199201442693120)/(10-3*Pi)^6 this is approximately to, 10, digits 150516.4157 147247015902844367465 6 and in terms of a=1/(10-3*Pi) this equals , - --------------------- a 977971151872 409917409335928875 5 939471791339925 4 3585475475 3 4776585 2 + ------------------ a - --------------- a + ---------- a - ------- a 1384527872 15049216 728 28 + 2475 a - 11 that in Maple notation is -147247015902844367465/977971151872*a^6+409917409335928875/1384527872*a^5-\ 939471791339925/15049216*a^4+3585475475/728*a^3-4776585/28*a^2+2475*a-11 To sum up, the limits of the scaled momentes (aka alpha coefficiets, from th\ e 3-rd to the, 12, -th are 1/2 / Pi \1/2 (6 Pi - 75/4) 3 |---------| 2 \10 - 3 Pi/ -189 Pi + 315 Pi + 884 [---------------------------------, -----------------------, 10 - 3 Pi 2 7 (10 - 3 Pi) / 2 105845\ 1/2 / Pi \1/2 |36 Pi + 75/2 Pi - ------| 3 |---------| \ 224 / \10 - 3 Pi/ -----------------------------------------------, 2 (10 - 3 Pi) 3 2 15 (-144144 Pi - 720720 Pi + 3013725 Pi + 2120320) ----------------------------------------------------, 3 16016 (10 - 3 Pi) / 3 2 103965 101897475\ 1/2 / Pi \1/2 |162 Pi + 6615/4 Pi - ------ Pi - ---------| 3 |---------| \ 32 9152 / \10 - 3 Pi/ ------------------------------------------------------------------, 3 ( 3 (10 - 3 Pi) 4 3 2 -488864376 Pi - 8147739600 Pi - 455885430 Pi + 86568885375 Pi / 4 + 32820007040) / (2586584 (10 - 3 Pi) ), / / 4 3 591867 2 461286225 188411947088175\ 1/2 |648 Pi + 15795 Pi + ------ Pi - --------- Pi - ---------------| 3 \ 16 2288 662165504 / / Pi \1/2 / 4 5 |---------| / (10 - 3 Pi) , 3 (-5551264502784 Pi \10 - 3 Pi/ / 4 3 2 - 185042150092800 Pi - 926881269874560 Pi + 1941922651536000 Pi / 5 / + 8623390317162525 Pi + 2206571720704000) / (7614903296 (10 - 3 Pi) ), | / \ 5 4 95982975 3 16485525 2 557660319717825 2430 Pi + 423225/4 Pi + -------- Pi - -------- Pi - --------------- Pi 112 416 60196864 2647571188398375\ 1/2 / Pi \1/2 / 5 - ----------------| 3 |---------| / (10 - 3 Pi) , 27 ( 325771264 / \10 - 3 Pi/ / 6 5 4 -4066404049483776 Pi - 223652222721607680 Pi - 2601772448089351680 Pi 3 2 - 4879832965588627200 Pi + 27707639928293288700 Pi / + 61080560747571248125 Pi + 11742199201442693120) / (13691596126208 / 6 (10 - 3 Pi) )] and in Maple notation: [(6*Pi-75/4)/(10-3*Pi)*3^(1/2)*(Pi/(10-3*Pi))^(1/2), 1/7*(-189*Pi^2+315*Pi+884) /(10-3*Pi)^2, (36*Pi^2+75/2*Pi-105845/224)/(10-3*Pi)^2*3^(1/2)*(Pi/(10-3*Pi))^( 1/2), 15/16016*(-144144*Pi^3-720720*Pi^2+3013725*Pi+2120320)/(10-3*Pi)^3, (162* Pi^3+6615/4*Pi^2-103965/32*Pi-101897475/9152)/(10-3*Pi)^3*3^(1/2)*(Pi/(10-3*Pi) )^(1/2), 3/2586584*(-488864376*Pi^4-8147739600*Pi^3-455885430*Pi^2+86568885375* Pi+32820007040)/(10-3*Pi)^4, (648*Pi^4+15795*Pi^3+591867/16*Pi^2-461286225/2288 *Pi-188411947088175/662165504)/(10-3*Pi)^4*3^(1/2)*(Pi/(10-3*Pi))^(1/2), 3/ 7614903296*(-5551264502784*Pi^5-185042150092800*Pi^4-926881269874560*Pi^3+ 1941922651536000*Pi^2+8623390317162525*Pi+2206571720704000)/(10-3*Pi)^5, (2430* Pi^5+423225/4*Pi^4+95982975/112*Pi^3-16485525/416*Pi^2-557660319717825/60196864 *Pi-2647571188398375/325771264)/(10-3*Pi)^5*3^(1/2)*(Pi/(10-3*Pi))^(1/2), 27/ 13691596126208*(-4066404049483776*Pi^6-223652222721607680*Pi^5-\ 2601772448089351680*Pi^4-4879832965588627200*Pi^3+27707639928293288700*Pi^2+ 61080560747571248125*Pi+11742199201442693120)/(10-3*Pi)^6] and in floating point: [.7005665208, 3.560394751, 7.256376376, 27.68549546, 90.01702180, 358.8086679, 1460.710269, 6498.233818, 30389.98955, 150516.4157] and in terms of a=1/(10-3*Pi) 1/2 (5 a - 8) (10 a - 1) 2 [-----------------------, -166/7 a + 45 a - 3, 4 2 1/2 (11755 a - 20720 a + 896) (10 a - 1) ----------------------------------------, 224 1264925 3 230625 2 - ------- a + ------ a - 225 a + 5, 1144 112 3 2 1/2 (22069225 a - 40195870 a + 3329040 a - 54912) (10 a - 1) -------------------------------------------------------------, 9152 68706293065 4 115208325 3 2 - ----------- a + --------- a - 59035/4 a + 630 a - 7, ( 1293292 1144 4 3 2 79090804803025 a - 147331855027840 a + 17521023391872 a - 599259781120 a 1/2 10394988504651375 5 + 5297324032) (10 a - 1) /662165504, - ----------------- a 3807451648 1746385189154325 4 1087630425 3 2 + ---------------- a - ---------- a + 468195/8 a - 1350 a + 9, 5 ( 331082752 1144 5 4 3 7039700396762925 a - 13370192732090140 a + 2028675442090240 a 2 1/2 - 104105864778240 a + 2000642775040 a - 11076222976) (10 a - 1) / 147247015902844367465 6 409917409335928875 5 5538111488, - --------------------- a + ------------------ a 977971151872 1384527872 939471791339925 4 3585475475 3 4776585 2 - --------------- a + ---------- a - ------- a + 2475 a - 11] 15049216 728 28 and in Maple notation: [1/4*(5*a-8)*(10*a-1)^(1/2), -166/7*a^2+45*a-3, 1/224*(11755*a^2-20720*a+896)*( 10*a-1)^(1/2), -1264925/1144*a^3+230625/112*a^2-225*a+5, 1/9152*(22069225*a^3-\ 40195870*a^2+3329040*a-54912)*(10*a-1)^(1/2), -68706293065/1293292*a^4+ 115208325/1144*a^3-59035/4*a^2+630*a-7, 1/662165504*(79090804803025*a^4-\ 147331855027840*a^3+17521023391872*a^2-599259781120*a+5297324032)*(10*a-1)^(1/2 ), -10394988504651375/3807451648*a^5+1746385189154325/331082752*a^4-1087630425/ 1144*a^3+468195/8*a^2-1350*a+9, 5/5538111488*(7039700396762925*a^5-\ 13370192732090140*a^4+2028675442090240*a^3-104105864778240*a^2+2000642775040*a-\ 11076222976)*(10*a-1)^(1/2), -147247015902844367465/977971151872*a^6+ 409917409335928875/1384527872*a^5-939471791339925/15049216*a^4+3585475475/728*a ^3-4776585/28*a^2+2475*a-11] -------------------------------------------------------------- This ends this article that took, 44212.319, seconds to generate. >