On the Asymptotic normality of the Democratic (Infinite Temperature) Ising Model For m by n strips for m between 1 and 4 By Shalosh B. Ekhad Suppose that you place mn people on an m by n torodial rectangle where at each location it is equally likely to have a man or a woman Consider the two random variables X:= The number of Homosexual neighboring pairs MINUS the number Heterosexual neigboring pairs Y:=The number of men MINUS the number of women then we have the following statistical facts For n by , 1, torodial rectangle The expectation of X is (of course), n The expectation of Y is (of course), 0 The covariance of X and Y is (of course), 0 The list-of-lists of mixed moments up to 7th moments is [[1, 0, n, 0, n*(3*n-2), 0, n*(15*n^2-30*n+16), 0], [0, 0, 2*n, 0, 4*n*(3*n-4), 0, 2*n*(45*n^2-150*n+136), 0], [n, 0, n*(n+4), 0, n*(3*n+52)*(n-2), 0, n*(15*n^ 3+510*n^2-2624*n+3424), 0], [0, 0, 2*n*(3*n+4), 0, 4*n*(3*n+20)*(3*n-8), 0, 2*n *(135*n^3+1170*n^2-10632*n+19264), 0], [n*(3*n-2), 0, n*(3*n^2+22*n+16), 0, n*( 9*n^3+276*n^2+148*n-3872), 0, n*(45*n^4+3120*n^3+3708*n^2-159968*n+411136), 0], [0, 0, 2*n*(15*n^2+30*n+16), 0, 4*n*(45*n^3+390*n^2-792*n-5824), 0, 2*n*(675*n^ 4+13500*n^3-48540*n^2-538320*n+2133376), 0], [n*(15*n^2-30*n+16), 0, n*(15*n^3+ 150*n^2+136*n+64), 0, n*(n+4)*(45*n^3+1860*n^2-1932*n-34976), 0, n*(225*n^5+ 23400*n^4+157980*n^3-1775040*n^2-5670464*n+43600384), 0], [0, 0, 2*n*(105*n^3+ 210*n^2+168*n+64), 0, 4*n*(315*n^4+3990*n^3-2856*n^2-99264*n-209920), 0, 2*n*( 4725*n^5+144900*n^4+2940*n^3-10471920*n^2-3595392*n+220801024), 0]] We will now find the mixed alpha coefficients to see whether the converge to those of the famous bi-variate Gaussian distribution The , 0, 0, mixed moment is 1 The , 0, 0, alpha coeff. is 1 that tends to, 1 as it should The , 0, 1, mixed moment is 0 The , 0, 1, alpha coeff. is 0 that tends to 0, as it should The , 0, 2, mixed moment is n The , 0, 2, alpha coeff. is 1 that tends to, 1 as it should The , 0, 3, mixed moment is 0 The , 0, 3, alpha coeff. is 0 that tends to 0, as it should The , 0, 4, mixed moment is n*(3*n-2) The , 0, 4, alpha coeff. is 3-2/n that tends to, 3 as it should The , 0, 5, mixed moment is 0 The , 0, 5, alpha coeff. is 0 that tends to 0, as it should The , 0, 6, mixed moment is n*(15*n^2-30*n+16) The , 0, 6, alpha coeff. is 15-30/n+16/n^2 that tends to, 15 as it should The , 0, 7, mixed moment is 0 The , 0, 7, alpha coeff. is 0 that tends to 0, as it should The , 1, 0, mixed moment is 0 The , 1, 0, alpha coeff. is 0 that tends to 0, as it should The , 1, 1, mixed moment is 0 The , 1, 1, alpha coeff. is 0 that tends to 0, as it should The , 1, 2, mixed moment is 2*n The , 1, 2, alpha coeff. is 2*(1/n)^(1/2) that tends to 0, as it should The , 1, 3, mixed moment is 0 The , 1, 3, alpha coeff. is 0 that tends to 0, as it should The , 1, 4, mixed moment is 4*n*(3*n-4) The , 1, 4, alpha coeff. is 12*(1/n)^(1/2)-16*(1/n)^(3/2) that tends to 0, as it should The , 1, 5, mixed moment is 0 The , 1, 5, alpha coeff. is 0 that tends to 0, as it should The , 1, 6, mixed moment is 2*n*(45*n^2-150*n+136) The , 1, 6, alpha coeff. is 90*(1/n)^(1/2)-300*(1/n)^(3/2)+272*(1/n)^(5/2) that tends to 0, as it should The , 1, 7, mixed moment is 0 The , 1, 7, alpha coeff. is 0 that tends to 0, as it should The , 2, 0, mixed moment is n The , 2, 0, alpha coeff. is 1 that tends to, 1 as it should The , 2, 1, mixed moment is 0 The , 2, 1, alpha coeff. is 0 that tends to 0, as it should The , 2, 2, mixed moment is n*(n+4) The , 2, 2, alpha coeff. is 1+4/n that tends to, 1 as it should The , 2, 3, mixed moment is 0 The , 2, 3, alpha coeff. is 0 that tends to 0, as it should The , 2, 4, mixed moment is n*(3*n+52)*(n-2) The , 2, 4, alpha coeff. is 3+46/n-104/n^2 that tends to, 3 as it should The , 2, 5, mixed moment is 0 The , 2, 5, alpha coeff. is 0 that tends to 0, as it should The , 2, 6, mixed moment is n*(15*n^3+510*n^2-2624*n+3424) The , 2, 6, alpha coeff. is 15+510/n-2624/n^2+3424/n^3 that tends to, 15 as it should The , 2, 7, mixed moment is 0 The , 2, 7, alpha coeff. is 0 that tends to 0, as it should The , 3, 0, mixed moment is 0 The , 3, 0, alpha coeff. is 0 that tends to 0, as it should The , 3, 1, mixed moment is 0 The , 3, 1, alpha coeff. is 0 that tends to 0, as it should The , 3, 2, mixed moment is 2*n*(3*n+4) The , 3, 2, alpha coeff. is 6*(1/n)^(1/2)+8*(1/n)^(3/2) that tends to 0, as it should The , 3, 3, mixed moment is 0 The , 3, 3, alpha coeff. is 0 that tends to 0, as it should The , 3, 4, mixed moment is 4*n*(3*n+20)*(3*n-8) The , 3, 4, alpha coeff. is 36*(1/n)^(1/2)+144*(1/n)^(3/2)-640*(1/n)^(5/2) that tends to 0, as it should The , 3, 5, mixed moment is 0 The , 3, 5, alpha coeff. is 0 that tends to 0, as it should The , 3, 6, mixed moment is 2*n*(135*n^3+1170*n^2-10632*n+19264) The , 3, 6, alpha coeff. is 270*(1/n)^(1/2)+2340*(1/n)^(3/2)-21264*(1/n)^(5/2)+38528*(1/n)^(7/2) that tends to 0, as it should The , 3, 7, mixed moment is 0 The , 3, 7, alpha coeff. is 0 that tends to 0, as it should The , 4, 0, mixed moment is n*(3*n-2) The , 4, 0, alpha coeff. is 3-2/n that tends to, 3 as it should The , 4, 1, mixed moment is 0 The , 4, 1, alpha coeff. is 0 that tends to 0, as it should The , 4, 2, mixed moment is n*(3*n^2+22*n+16) The , 4, 2, alpha coeff. is 3+22/n+16/n^2 that tends to, 3 as it should The , 4, 3, mixed moment is 0 The , 4, 3, alpha coeff. is 0 that tends to 0, as it should The , 4, 4, mixed moment is n*(9*n^3+276*n^2+148*n-3872) The , 4, 4, alpha coeff. is 9+276/n+148/n^2-3872/n^3 that tends to, 9 as it should The , 4, 5, mixed moment is 0 The , 4, 5, alpha coeff. is 0 that tends to 0, as it should The , 4, 6, mixed moment is n*(45*n^4+3120*n^3+3708*n^2-159968*n+411136) The , 4, 6, alpha coeff. is 45+3120/n+3708/n^2-159968/n^3+411136/n^4 that tends to, 45 as it should The , 4, 7, mixed moment is 0 The , 4, 7, alpha coeff. is 0 that tends to 0, as it should The , 5, 0, mixed moment is 0 The , 5, 0, alpha coeff. is 0 that tends to 0, as it should The , 5, 1, mixed moment is 0 The , 5, 1, alpha coeff. is 0 that tends to 0, as it should The , 5, 2, mixed moment is 2*n*(15*n^2+30*n+16) The , 5, 2, alpha coeff. is 30*(1/n)^(1/2)+60*(1/n)^(3/2)+32*(1/n)^(5/2) that tends to 0, as it should The , 5, 3, mixed moment is 0 The , 5, 3, alpha coeff. is 0 that tends to 0, as it should The , 5, 4, mixed moment is 4*n*(45*n^3+390*n^2-792*n-5824) The , 5, 4, alpha coeff. is 180*(1/n)^(1/2)+1560*(1/n)^(3/2)-3168*(1/n)^(5/2)-23296*(1/n)^(7/2) that tends to 0, as it should The , 5, 5, mixed moment is 0 The , 5, 5, alpha coeff. is 0 that tends to 0, as it should The , 5, 6, mixed moment is 2*n*(675*n^4+13500*n^3-48540*n^2-538320*n+2133376) The , 5, 6, alpha coeff. is 1350*(1/n)^(1/2)+27000*(1/n)^(3/2)-97080*(1/n)^(5/2)-1076640*(1/n)^(7/2)+ 4266752*(1/n)^(9/2) that tends to 0, as it should The , 5, 7, mixed moment is 0 The , 5, 7, alpha coeff. is 0 that tends to 0, as it should The , 6, 0, mixed moment is n*(15*n^2-30*n+16) The , 6, 0, alpha coeff. is 15-30/n+16/n^2 that tends to, 15 as it should The , 6, 1, mixed moment is 0 The , 6, 1, alpha coeff. is 0 that tends to 0, as it should The , 6, 2, mixed moment is n*(15*n^3+150*n^2+136*n+64) The , 6, 2, alpha coeff. is 15+150/n+136/n^2+64/n^3 that tends to, 15 as it should The , 6, 3, mixed moment is 0 The , 6, 3, alpha coeff. is 0 that tends to 0, as it should The , 6, 4, mixed moment is n*(n+4)*(45*n^3+1860*n^2-1932*n-34976) The , 6, 4, alpha coeff. is 45+2040/n+5508/n^2-42704/n^3-139904/n^4 that tends to, 45 as it should The , 6, 5, mixed moment is 0 The , 6, 5, alpha coeff. is 0 that tends to 0, as it should The , 6, 6, mixed moment is n*(225*n^5+23400*n^4+157980*n^3-1775040*n^2-5670464*n+43600384) The , 6, 6, alpha coeff. is 225+23400/n+157980/n^2-1775040/n^3-5670464/n^4+43600384/n^5 that tends to, 225 as it should The , 6, 7, mixed moment is 0 The , 6, 7, alpha coeff. is 0 that tends to 0, as it should The , 7, 0, mixed moment is 0 The , 7, 0, alpha coeff. is 0 that tends to 0, as it should The , 7, 1, mixed moment is 0 The , 7, 1, alpha coeff. is 0 that tends to 0, as it should The , 7, 2, mixed moment is 2*n*(105*n^3+210*n^2+168*n+64) The , 7, 2, alpha coeff. is 210*(1/n)^(1/2)+420*(1/n)^(3/2)+336*(1/n)^(5/2)+128*(1/n)^(7/2) that tends to 0, as it should The , 7, 3, mixed moment is 0 The , 7, 3, alpha coeff. is 0 that tends to 0, as it should The , 7, 4, mixed moment is 4*n*(315*n^4+3990*n^3-2856*n^2-99264*n-209920) The , 7, 4, alpha coeff. is 1260*(1/n)^(1/2)+15960*(1/n)^(3/2)-11424*(1/n)^(5/2)-397056*(1/n)^(7/2)-839680* (1/n)^(9/2) that tends to 0, as it should The , 7, 5, mixed moment is 0 The , 7, 5, alpha coeff. is 0 that tends to 0, as it should The , 7, 6, mixed moment is 2*n*(4725*n^5+144900*n^4+2940*n^3-10471920*n^2-3595392*n+220801024) The , 7, 6, alpha coeff. is 9450*(1/n)^(1/2)+289800*(1/n)^(3/2)+5880*(1/n)^(5/2)-20943840*(1/n)^(7/2)-\ 7190784*(1/n)^(9/2)+441602048*(1/n)^(11/2) that tends to 0, as it should The , 7, 7, mixed moment is 0 The , 7, 7, alpha coeff. is 0 that tends to 0, as it should For n by , 2, torodial rectangle The expectation of X is (of course), 0 The expectation of Y is (of course), 0 The covariance of X and Y is (of course), 0 The list-of-lists of mixed moments up to 7th moments is [[1, 0, 2*n, 0, 4*n*(3*n-1), 0, 8*n*(15*n^2-15*n+4), 0], [0, 0, 8*n, 0, 32*n*(3 *n-2), 0, 32*n*(45*n^2-75*n+34), 0], [6*n, 0, 4*n*(3*n+10), 0, 8*n*(9*n^2+105*n -118), 0, 16*n*(45*n^3+1125*n^2-2868*n+1900), 0], [0, 0, 16*n*(9*n+17), 0, 64*n *(27*n^2+123*n-232), 0, 64*n*(405*n^3+3510*n^2-14019*n+12908), 0], [12*n*(9*n+5 ), 0, 8*n*(9*n+53)*(3*n+4), 0, 16*n*(81*n^3+1962*n^2+4197*n-14756), 0, 32*n*( 405*n^4+20880*n^3+76743*n^2-551820*n+686048), 0], [0, 0, 32*n*(135*n^2+585*n+ 274), 0, 128*n*(405*n^3+4185*n^2+1992*n-29708), 0, 128*n*(6075*n^4+118800*n^3+ 87345*n^2-2672760*n+4525516), 0], [24*n*(135*n^2+225*n-236), 0, 16*n*(405*n^3+ 4725*n^2+11082*n+1960), 0, 32*n*(1215*n^4+45360*n^3+258831*n^2-269670*n-1940632 ), 0, 64*n*(6075*n^5+477900*n^4+4869225*n^3-9040500*n^2-99171492*n+239148880), 0], [0, 0, 64*n*(2835*n^3+20790*n^2+21231*n-7972), 0, 256*n*(8505*n^4+141750*n^ 3+403893*n^2-1415208*n-3921808), 0, 256*n*(127575*n^5+3954825*n^4+20821185*n^3-\ 115516125*n^2-411093996*n+1582619072), 0]] We will now find the mixed alpha coefficients to see whether the converge to those of the famous bi-variate Gaussian distribution The , 0, 0, mixed moment is 1 The , 0, 0, alpha coeff. is 1 that tends to, 1 as it should The , 0, 1, mixed moment is 0 The , 0, 1, alpha coeff. is 0 that tends to 0, as it should The , 0, 2, mixed moment is 2*n The , 0, 2, alpha coeff. is 1 that tends to, 1 as it should The , 0, 3, mixed moment is 0 The , 0, 3, alpha coeff. is 0 that tends to 0, as it should The , 0, 4, mixed moment is 4*n*(3*n-1) The , 0, 4, alpha coeff. is 3-1/n that tends to, 3 as it should The , 0, 5, mixed moment is 0 The , 0, 5, alpha coeff. is 0 that tends to 0, as it should The , 0, 6, mixed moment is 8*n*(15*n^2-15*n+4) The , 0, 6, alpha coeff. is 15-15/n+4/n^2 that tends to, 15 as it should The , 0, 7, mixed moment is 0 The , 0, 7, alpha coeff. is 0 that tends to 0, as it should The , 1, 0, mixed moment is 0 The , 1, 0, alpha coeff. is 0 that tends to 0, as it should The , 1, 1, mixed moment is 0 The , 1, 1, alpha coeff. is 0 that tends to 0, as it should The , 1, 2, mixed moment is 8*n The , 1, 2, alpha coeff. is 2/3*6^(1/2)*(1/n)^(1/2) that tends to 0, as it should The , 1, 3, mixed moment is 0 The , 1, 3, alpha coeff. is 0 that tends to 0, as it should The , 1, 4, mixed moment is 32*n*(3*n-2) The , 1, 4, alpha coeff. is 4*6^(1/2)*(1/n)^(1/2)-8/3*6^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 1, 5, mixed moment is 0 The , 1, 5, alpha coeff. is 0 that tends to 0, as it should The , 1, 6, mixed moment is 32*n*(45*n^2-75*n+34) The , 1, 6, alpha coeff. is 30*6^(1/2)*(1/n)^(1/2)-50*6^(1/2)*(1/n)^(3/2)+68/3*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 1, 7, mixed moment is 0 The , 1, 7, alpha coeff. is 0 that tends to 0, as it should The , 2, 0, mixed moment is 6*n The , 2, 0, alpha coeff. is 1 that tends to, 1 as it should The , 2, 1, mixed moment is 0 The , 2, 1, alpha coeff. is 0 that tends to 0, as it should The , 2, 2, mixed moment is 4*n*(3*n+10) The , 2, 2, alpha coeff. is 1+10/3/n that tends to, 1 as it should The , 2, 3, mixed moment is 0 The , 2, 3, alpha coeff. is 0 that tends to 0, as it should The , 2, 4, mixed moment is 8*n*(9*n^2+105*n-118) The , 2, 4, alpha coeff. is 3+35/n-118/3/n^2 that tends to, 3 as it should The , 2, 5, mixed moment is 0 The , 2, 5, alpha coeff. is 0 that tends to 0, as it should The , 2, 6, mixed moment is 16*n*(45*n^3+1125*n^2-2868*n+1900) The , 2, 6, alpha coeff. is 15+375/n-956/n^2+1900/3/n^3 that tends to, 15 as it should The , 2, 7, mixed moment is 0 The , 2, 7, alpha coeff. is 0 that tends to 0, as it should The , 3, 0, mixed moment is 0 The , 3, 0, alpha coeff. is 0 that tends to 0, as it should The , 3, 1, mixed moment is 0 The , 3, 1, alpha coeff. is 0 that tends to 0, as it should The , 3, 2, mixed moment is 16*n*(9*n+17) The , 3, 2, alpha coeff. is 2*6^(1/2)*(1/n)^(1/2)+34/9*6^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 3, 3, mixed moment is 0 The , 3, 3, alpha coeff. is 0 that tends to 0, as it should The , 3, 4, mixed moment is 64*n*(27*n^2+123*n-232) The , 3, 4, alpha coeff. is 12*6^(1/2)*(1/n)^(1/2)+164/3*6^(1/2)*(1/n)^(3/2)-928/9*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 3, 5, mixed moment is 0 The , 3, 5, alpha coeff. is 0 that tends to 0, as it should The , 3, 6, mixed moment is 64*n*(405*n^3+3510*n^2-14019*n+12908) The , 3, 6, alpha coeff. is 90*6^(1/2)*(1/n)^(1/2)+780*6^(1/2)*(1/n)^(3/2)-9346/3*6^(1/2)*(1/n)^(5/2)+25816 /9*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 3, 7, mixed moment is 0 The , 3, 7, alpha coeff. is 0 that tends to 0, as it should The , 4, 0, mixed moment is 12*n*(9*n+5) The , 4, 0, alpha coeff. is 3+5/3/n that tends to, 3 as it should The , 4, 1, mixed moment is 0 The , 4, 1, alpha coeff. is 0 that tends to 0, as it should The , 4, 2, mixed moment is 8*n*(9*n+53)*(3*n+4) The , 4, 2, alpha coeff. is 3+65/3/n+212/9/n^2 that tends to, 3 as it should The , 4, 3, mixed moment is 0 The , 4, 3, alpha coeff. is 0 that tends to 0, as it should The , 4, 4, mixed moment is 16*n*(81*n^3+1962*n^2+4197*n-14756) The , 4, 4, alpha coeff. is 9+218/n+1399/3/n^2-14756/9/n^3 that tends to, 9 as it should The , 4, 5, mixed moment is 0 The , 4, 5, alpha coeff. is 0 that tends to 0, as it should The , 4, 6, mixed moment is 32*n*(405*n^4+20880*n^3+76743*n^2-551820*n+686048) The , 4, 6, alpha coeff. is 45+2320/n+8527/n^2-183940/3/n^3+686048/9/n^4 that tends to, 45 as it should The , 4, 7, mixed moment is 0 The , 4, 7, alpha coeff. is 0 that tends to 0, as it should The , 5, 0, mixed moment is 0 The , 5, 0, alpha coeff. is 0 that tends to 0, as it should The , 5, 1, mixed moment is 0 The , 5, 1, alpha coeff. is 0 that tends to 0, as it should The , 5, 2, mixed moment is 32*n*(135*n^2+585*n+274) The , 5, 2, alpha coeff. is 10*6^(1/2)*(1/n)^(1/2)+130/3*6^(1/2)*(1/n)^(3/2)+548/27*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 5, 3, mixed moment is 0 The , 5, 3, alpha coeff. is 0 that tends to 0, as it should The , 5, 4, mixed moment is 128*n*(405*n^3+4185*n^2+1992*n-29708) The , 5, 4, alpha coeff. is 60*6^(1/2)*(1/n)^(1/2)+620*6^(1/2)*(1/n)^(3/2)+2656/9*6^(1/2)*(1/n)^(5/2)-\ 118832/27*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 5, 5, mixed moment is 0 The , 5, 5, alpha coeff. is 0 that tends to 0, as it should The , 5, 6, mixed moment is 128*n*(6075*n^4+118800*n^3+87345*n^2-2672760*n+4525516) The , 5, 6, alpha coeff. is 450*6^(1/2)*(1/n)^(1/2)+8800*6^(1/2)*(1/n)^(3/2)+6470*6^(1/2)*(1/n)^(5/2)-\ 1781840/9*6^(1/2)*(1/n)^(7/2)+9051032/27*6^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 5, 7, mixed moment is 0 The , 5, 7, alpha coeff. is 0 that tends to 0, as it should The , 6, 0, mixed moment is 24*n*(135*n^2+225*n-236) The , 6, 0, alpha coeff. is 15+25/n-236/9/n^2 that tends to, 15 as it should The , 6, 1, mixed moment is 0 The , 6, 1, alpha coeff. is 0 that tends to 0, as it should The , 6, 2, mixed moment is 16*n*(405*n^3+4725*n^2+11082*n+1960) The , 6, 2, alpha coeff. is 15+175/n+3694/9/n^2+1960/27/n^3 that tends to, 15 as it should The , 6, 3, mixed moment is 0 The , 6, 3, alpha coeff. is 0 that tends to 0, as it should The , 6, 4, mixed moment is 32*n*(1215*n^4+45360*n^3+258831*n^2-269670*n-1940632) The , 6, 4, alpha coeff. is 45+1680/n+28759/3/n^2-89890/9/n^3-1940632/27/n^4 that tends to, 45 as it should The , 6, 5, mixed moment is 0 The , 6, 5, alpha coeff. is 0 that tends to 0, as it should The , 6, 6, mixed moment is 64*n*(6075*n^5+477900*n^4+4869225*n^3-9040500*n^2-99171492*n+239148880) The , 6, 6, alpha coeff. is 225+17700/n+541025/3/n^2-1004500/3/n^3-33057164/9/n^4+239148880/27/n^5 that tends to, 225 as it should The , 6, 7, mixed moment is 0 The , 6, 7, alpha coeff. is 0 that tends to 0, as it should The , 7, 0, mixed moment is 0 The , 7, 0, alpha coeff. is 0 that tends to 0, as it should The , 7, 1, mixed moment is 0 The , 7, 1, alpha coeff. is 0 that tends to 0, as it should The , 7, 2, mixed moment is 64*n*(2835*n^3+20790*n^2+21231*n-7972) The , 7, 2, alpha coeff. is 70*6^(1/2)*(1/n)^(1/2)+1540/3*6^(1/2)*(1/n)^(3/2)+4718/9*6^(1/2)*(1/n)^(5/2)-\ 15944/81*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 7, 3, mixed moment is 0 The , 7, 3, alpha coeff. is 0 that tends to 0, as it should The , 7, 4, mixed moment is 256*n*(8505*n^4+141750*n^3+403893*n^2-1415208*n-3921808) The , 7, 4, alpha coeff. is 420*6^(1/2)*(1/n)^(1/2)+7000*6^(1/2)*(1/n)^(3/2)+59836/3*6^(1/2)*(1/n)^(5/2)-\ 1886944/27*6^(1/2)*(1/n)^(7/2)-15687232/81*6^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 7, 5, mixed moment is 0 The , 7, 5, alpha coeff. is 0 that tends to 0, as it should The , 7, 6, mixed moment is 256*n*(127575*n^5+3954825*n^4+20821185*n^3-115516125*n^2-411093996*n+1582619072 ) The , 7, 6, alpha coeff. is 3150*6^(1/2)*(1/n)^(1/2)+97650*6^(1/2)*(1/n)^(3/2)+1542310/3*6^(1/2)*(1/n)^(5/2 )-2852250*6^(1/2)*(1/n)^(7/2)-274062664/27*6^(1/2)*(1/n)^(9/2)+3165238144/81*6^ (1/2)*(1/n)^(11/2) that tends to 0, as it should The , 7, 7, mixed moment is 0 The , 7, 7, alpha coeff. is 0 that tends to 0, as it should For n by , 3, torodial rectangle The expectation of X is (of course), 0 The expectation of Y is (of course), 0 The covariance of X and Y is (of course), 0 The list-of-lists of mixed moments up to 7th moments is [[1, 0, 3*n, 0, 3*n*(9*n-2), 0, 3*n*(135*n^2-90*n+16), 0], [0, 0, 12*n, 0, 24*n *(9*n-4), 0, 12*n*(405*n^2-450*n+136), 0], [6*n, 0, 18*n*(n+4), 0, 18*n*(9*n^2+ 118*n-88), 0, 18*n*(135*n^3+3690*n^2-6224*n+2784), 0], [6*n, 0, 6*n*(39*n+98), 0, 6*n*(675*n^2+4062*n-4744), 0, 6*n*(14985*n^3+165780*n^2-411756*n+254528), 0] , [12*n*(9*n+5), 0, 36*n*(9*n^2+85*n+160), 0, 36*n*(81*n^3+2331*n^2+8518*n-\ 15408), 0, 36*n*(1215*n^4+71145*n^3+432414*n^2-1604960*n+1318400), 0], [120*n*( 3*n+5), 0, 24*n*(315*n^2+1875*n+2558), 0, 24*n*(5265*n^3+71865*n^2+161814*n-\ 480544), 0, 24*n*(115425*n^4+2849175*n^3+9793080*n^2-59048700*n+63081488), 0], [24*n*(135*n^2+240*n+49), 0, 216*n*(45*n^3+740*n^2+3243*n+3212), 0, 72*n*(1215* n^4+56970*n^3+500721*n^2+582822*n-3522632), 0, 72*n*(18225*n^5+1696950*n^4+ 24902775*n^3+38228730*n^2-500220096*n+690234656), 0], [168*n*(135*n^2+525*n-98) , 0, 24*n*(14175*n^3+146475*n^2+464016*n+335612), 0, 24*n*(229635*n^4+5162535*n ^3+31540320*n^2+4698948*n-244599856), 0, 24*n*(4975425*n^5+197953875*n^4+ 1929483990*n^3-367111980*n^2-39016329864*n+70067421632), 0]] We will now find the mixed alpha coefficients to see whether the converge to those of the famous bi-variate Gaussian distribution The , 0, 0, mixed moment is 1 The , 0, 0, alpha coeff. is 1 that tends to, 1 as it should The , 0, 1, mixed moment is 0 The , 0, 1, alpha coeff. is 0 that tends to 0, as it should The , 0, 2, mixed moment is 3*n The , 0, 2, alpha coeff. is 1 that tends to, 1 as it should The , 0, 3, mixed moment is 0 The , 0, 3, alpha coeff. is 0 that tends to 0, as it should The , 0, 4, mixed moment is 3*n*(9*n-2) The , 0, 4, alpha coeff. is 3-2/3/n that tends to, 3 as it should The , 0, 5, mixed moment is 0 The , 0, 5, alpha coeff. is 0 that tends to 0, as it should The , 0, 6, mixed moment is 3*n*(135*n^2-90*n+16) The , 0, 6, alpha coeff. is 15-10/n+16/9/n^2 that tends to, 15 as it should The , 0, 7, mixed moment is 0 The , 0, 7, alpha coeff. is 0 that tends to 0, as it should The , 1, 0, mixed moment is 0 The , 1, 0, alpha coeff. is 0 that tends to 0, as it should The , 1, 1, mixed moment is 0 The , 1, 1, alpha coeff. is 0 that tends to 0, as it should The , 1, 2, mixed moment is 12*n The , 1, 2, alpha coeff. is 2/3*6^(1/2)*(1/n)^(1/2) that tends to 0, as it should The , 1, 3, mixed moment is 0 The , 1, 3, alpha coeff. is 0 that tends to 0, as it should The , 1, 4, mixed moment is 24*n*(9*n-4) The , 1, 4, alpha coeff. is 4*6^(1/2)*(1/n)^(1/2)-16/9*6^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 1, 5, mixed moment is 0 The , 1, 5, alpha coeff. is 0 that tends to 0, as it should The , 1, 6, mixed moment is 12*n*(405*n^2-450*n+136) The , 1, 6, alpha coeff. is 30*6^(1/2)*(1/n)^(1/2)-100/3*6^(1/2)*(1/n)^(3/2)+272/27*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 1, 7, mixed moment is 0 The , 1, 7, alpha coeff. is 0 that tends to 0, as it should The , 2, 0, mixed moment is 6*n The , 2, 0, alpha coeff. is 1 that tends to, 1 as it should The , 2, 1, mixed moment is 0 The , 2, 1, alpha coeff. is 0 that tends to 0, as it should The , 2, 2, mixed moment is 18*n*(n+4) The , 2, 2, alpha coeff. is 1+4/n that tends to, 1 as it should The , 2, 3, mixed moment is 0 The , 2, 3, alpha coeff. is 0 that tends to 0, as it should The , 2, 4, mixed moment is 18*n*(9*n^2+118*n-88) The , 2, 4, alpha coeff. is 3+118/3/n-88/3/n^2 that tends to, 3 as it should The , 2, 5, mixed moment is 0 The , 2, 5, alpha coeff. is 0 that tends to 0, as it should The , 2, 6, mixed moment is 18*n*(135*n^3+3690*n^2-6224*n+2784) The , 2, 6, alpha coeff. is 15+410/n-6224/9/n^2+928/3/n^3 that tends to, 15 as it should The , 2, 7, mixed moment is 0 The , 2, 7, alpha coeff. is 0 that tends to 0, as it should The , 3, 0, mixed moment is 6*n The , 3, 0, alpha coeff. is 1/6*6^(1/2)*(1/n)^(1/2) that tends to 0, as it should The , 3, 1, mixed moment is 0 The , 3, 1, alpha coeff. is 0 that tends to 0, as it should The , 3, 2, mixed moment is 6*n*(39*n+98) The , 3, 2, alpha coeff. is 13/6*6^(1/2)*(1/n)^(1/2)+49/9*6^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 3, 3, mixed moment is 0 The , 3, 3, alpha coeff. is 0 that tends to 0, as it should The , 3, 4, mixed moment is 6*n*(675*n^2+4062*n-4744) The , 3, 4, alpha coeff. is 25/2*6^(1/2)*(1/n)^(1/2)+677/9*6^(1/2)*(1/n)^(3/2)-2372/27*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 3, 5, mixed moment is 0 The , 3, 5, alpha coeff. is 0 that tends to 0, as it should The , 3, 6, mixed moment is 6*n*(14985*n^3+165780*n^2-411756*n+254528) The , 3, 6, alpha coeff. is 185/2*6^(1/2)*(1/n)^(1/2)+3070/3*6^(1/2)*(1/n)^(3/2)-68626/27*6^(1/2)*(1/n)^(5/ 2)+127264/81*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 3, 7, mixed moment is 0 The , 3, 7, alpha coeff. is 0 that tends to 0, as it should The , 4, 0, mixed moment is 12*n*(9*n+5) The , 4, 0, alpha coeff. is 3+5/3/n that tends to, 3 as it should The , 4, 1, mixed moment is 0 The , 4, 1, alpha coeff. is 0 that tends to 0, as it should The , 4, 2, mixed moment is 36*n*(9*n^2+85*n+160) The , 4, 2, alpha coeff. is 3+85/3/n+160/3/n^2 that tends to, 3 as it should The , 4, 3, mixed moment is 0 The , 4, 3, alpha coeff. is 0 that tends to 0, as it should The , 4, 4, mixed moment is 36*n*(81*n^3+2331*n^2+8518*n-15408) The , 4, 4, alpha coeff. is 9+259/n+8518/9/n^2-1712/n^3 that tends to, 9 as it should The , 4, 5, mixed moment is 0 The , 4, 5, alpha coeff. is 0 that tends to 0, as it should The , 4, 6, mixed moment is 36*n*(1215*n^4+71145*n^3+432414*n^2-1604960*n+1318400) The , 4, 6, alpha coeff. is 45+2635/n+48046/3/n^2-1604960/27/n^3+1318400/27/n^4 that tends to, 45 as it should The , 4, 7, mixed moment is 0 The , 4, 7, alpha coeff. is 0 that tends to 0, as it should The , 5, 0, mixed moment is 120*n*(3*n+5) The , 5, 0, alpha coeff. is 5/3*6^(1/2)*(1/n)^(1/2)+25/9*6^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 5, 1, mixed moment is 0 The , 5, 1, alpha coeff. is 0 that tends to 0, as it should The , 5, 2, mixed moment is 24*n*(315*n^2+1875*n+2558) The , 5, 2, alpha coeff. is 35/3*6^(1/2)*(1/n)^(1/2)+625/9*6^(1/2)*(1/n)^(3/2)+2558/27*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 5, 3, mixed moment is 0 The , 5, 3, alpha coeff. is 0 that tends to 0, as it should The , 5, 4, mixed moment is 24*n*(5265*n^3+71865*n^2+161814*n-480544) The , 5, 4, alpha coeff. is 65*6^(1/2)*(1/n)^(1/2)+7985/9*6^(1/2)*(1/n)^(3/2)+53938/27*6^(1/2)*(1/n)^(5/2)-\ 480544/81*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 5, 5, mixed moment is 0 The , 5, 5, alpha coeff. is 0 that tends to 0, as it should The , 5, 6, mixed moment is 24*n*(115425*n^4+2849175*n^3+9793080*n^2-59048700*n+63081488) The , 5, 6, alpha coeff. is 475*6^(1/2)*(1/n)^(1/2)+11725*6^(1/2)*(1/n)^(3/2)+1088120/27*6^(1/2)*(1/n)^(5/2 )-19682900/81*6^(1/2)*(1/n)^(7/2)+63081488/243*6^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 5, 7, mixed moment is 0 The , 5, 7, alpha coeff. is 0 that tends to 0, as it should The , 6, 0, mixed moment is 24*n*(135*n^2+240*n+49) The , 6, 0, alpha coeff. is 15+80/3/n+49/9/n^2 that tends to, 15 as it should The , 6, 1, mixed moment is 0 The , 6, 1, alpha coeff. is 0 that tends to 0, as it should The , 6, 2, mixed moment is 216*n*(45*n^3+740*n^2+3243*n+3212) The , 6, 2, alpha coeff. is 15+740/3/n+1081/n^2+3212/3/n^3 that tends to, 15 as it should The , 6, 3, mixed moment is 0 The , 6, 3, alpha coeff. is 0 that tends to 0, as it should The , 6, 4, mixed moment is 72*n*(1215*n^4+56970*n^3+500721*n^2+582822*n-3522632) The , 6, 4, alpha coeff. is 45+2110/n+166907/9/n^2+21586/n^3-3522632/27/n^4 that tends to, 45 as it should The , 6, 5, mixed moment is 0 The , 6, 5, alpha coeff. is 0 that tends to 0, as it should The , 6, 6, mixed moment is 72*n*(18225*n^5+1696950*n^4+24902775*n^3+38228730*n^2-500220096*n+690234656) The , 6, 6, alpha coeff. is 225+20950/n+922325/3/n^2+12742910/27/n^3-166740032/27/n^4+690234656/81/n^5 that tends to, 225 as it should The , 6, 7, mixed moment is 0 The , 6, 7, alpha coeff. is 0 that tends to 0, as it should The , 7, 0, mixed moment is 168*n*(135*n^2+525*n-98) The , 7, 0, alpha coeff. is 35/2*6^(1/2)*(1/n)^(1/2)+1225/18*6^(1/2)*(1/n)^(3/2)-343/27*6^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 7, 1, mixed moment is 0 The , 7, 1, alpha coeff. is 0 that tends to 0, as it should The , 7, 2, mixed moment is 24*n*(14175*n^3+146475*n^2+464016*n+335612) The , 7, 2, alpha coeff. is 175/2*6^(1/2)*(1/n)^(1/2)+5425/6*6^(1/2)*(1/n)^(3/2)+77336/27*6^(1/2)*(1/n)^(5/ 2)+167806/81*6^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 7, 3, mixed moment is 0 The , 7, 3, alpha coeff. is 0 that tends to 0, as it should The , 7, 4, mixed moment is 24*n*(229635*n^4+5162535*n^3+31540320*n^2+4698948*n-244599856) The , 7, 4, alpha coeff. is 945/2*6^(1/2)*(1/n)^(1/2)+21245/2*6^(1/2)*(1/n)^(3/2)+584080/9*6^(1/2)*(1/n)^(5 /2)+783158/81*6^(1/2)*(1/n)^(7/2)-122299928/243*6^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 7, 5, mixed moment is 0 The , 7, 5, alpha coeff. is 0 that tends to 0, as it should The , 7, 6, mixed moment is 24*n*(4975425*n^5+197953875*n^4+1929483990*n^3-367111980*n^2-39016329864*n+ 70067421632) The , 7, 6, alpha coeff. is 6825/2*6^(1/2)*(1/n)^(1/2)+814625/6*6^(1/2)*(1/n)^(3/2)+11910395/9*6^(1/2)*(1/n )^(5/2)-6798370/27*6^(1/2)*(1/n)^(7/2)-6502721644/243*6^(1/2)*(1/n)^(9/2)+ 35033710816/729*6^(1/2)*(1/n)^(11/2) that tends to 0, as it should The , 7, 7, mixed moment is 0 The , 7, 7, alpha coeff. is 0 that tends to 0, as it should For n by , 4, torodial rectangle The expectation of X is (of course), 0 The expectation of Y is (of course), 0 The covariance of X and Y is (of course), 0 The list-of-lists of mixed moments up to 7th moments is [[1, 0, 4*n, 0, 8*n*(6*n-1), 0, 32*n*(30*n^2-15*n+2), 0], [0, 0, 16*n, 0, 128*n *(3*n-1), 0, 128*n*(90*n^2-75*n+17), 0], [8*n, 0, 32*n*(n+3), 0, 64*n*(6*n^2+59 *n-33), 0, 256*n*(30*n^3+615*n^2-778*n+261), 0], [0, 0, 64*n*(6*n+13), 0, 512*n *(18*n^2+87*n-76), 0, 512*n*(540*n^3+4680*n^2-8703*n+4046), 0], [8*n*(24*n+13), 0, 32*n*(24*n^2+157*n+270), 0, 64*n*(144*n^3+2934*n^2+9227*n-12270), 0, 256*n*( 720*n^4+30270*n^3+152313*n^2-417706*n+258240), 0], [0, 0, 128*n*(120*n^2+585*n+ 857), 0, 1024*n*(360*n^3+3795*n^2+8196*n-16967), 0, 1024*n*(10800*n^4+202050*n^ 3+627495*n^2-2673780*n+2147119), 0], [64*n*(120*n^2+195*n+152), 0, 256*n*(120*n ^3+1275*n^2+4787*n+5916), 0, 512*n*(720*n^4+22650*n^3+162897*n^2+230683*n-\ 810306), 0, 2048*n*(3600*n^5+230850*n^4+2692425*n^3+4684230*n^2-36181211*n+ 37264692), 0], [0, 0, 512*n*(1680*n^3+13650*n^2+44037*n+45446), 0, 4096*n*(5040 *n^4+84630*n^3+456771*n^2+327984*n-2591402), 0, 4096*n*(151200*n^5+4428900*n^4+ 36219540*n^3+18986385*n^2-512165706*n+674894632), 0]] We will now find the mixed alpha coefficients to see whether the converge to those of the famous bi-variate Gaussian distribution The , 0, 0, mixed moment is 1 The , 0, 0, alpha coeff. is 1 that tends to, 1 as it should The , 0, 1, mixed moment is 0 The , 0, 1, alpha coeff. is 0 that tends to 0, as it should The , 0, 2, mixed moment is 4*n The , 0, 2, alpha coeff. is 1 that tends to, 1 as it should The , 0, 3, mixed moment is 0 The , 0, 3, alpha coeff. is 0 that tends to 0, as it should The , 0, 4, mixed moment is 8*n*(6*n-1) The , 0, 4, alpha coeff. is 3-1/2/n that tends to, 3 as it should The , 0, 5, mixed moment is 0 The , 0, 5, alpha coeff. is 0 that tends to 0, as it should The , 0, 6, mixed moment is 32*n*(30*n^2-15*n+2) The , 0, 6, alpha coeff. is 15-15/2/n+1/n^2 that tends to, 15 as it should The , 0, 7, mixed moment is 0 The , 0, 7, alpha coeff. is 0 that tends to 0, as it should The , 1, 0, mixed moment is 0 The , 1, 0, alpha coeff. is 0 that tends to 0, as it should The , 1, 1, mixed moment is 0 The , 1, 1, alpha coeff. is 0 that tends to 0, as it should The , 1, 2, mixed moment is 16*n The , 1, 2, alpha coeff. is 1/2*8^(1/2)*(1/n)^(1/2) that tends to 0, as it should The , 1, 3, mixed moment is 0 The , 1, 3, alpha coeff. is 0 that tends to 0, as it should The , 1, 4, mixed moment is 128*n*(3*n-1) The , 1, 4, alpha coeff. is 3*8^(1/2)*(1/n)^(1/2)-8^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 1, 5, mixed moment is 0 The , 1, 5, alpha coeff. is 0 that tends to 0, as it should The , 1, 6, mixed moment is 128*n*(90*n^2-75*n+17) The , 1, 6, alpha coeff. is 45/2*8^(1/2)*(1/n)^(1/2)-75/4*8^(1/2)*(1/n)^(3/2)+17/4*8^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 1, 7, mixed moment is 0 The , 1, 7, alpha coeff. is 0 that tends to 0, as it should The , 2, 0, mixed moment is 8*n The , 2, 0, alpha coeff. is 1 that tends to, 1 as it should The , 2, 1, mixed moment is 0 The , 2, 1, alpha coeff. is 0 that tends to 0, as it should The , 2, 2, mixed moment is 32*n*(n+3) The , 2, 2, alpha coeff. is 1+3/n that tends to, 1 as it should The , 2, 3, mixed moment is 0 The , 2, 3, alpha coeff. is 0 that tends to 0, as it should The , 2, 4, mixed moment is 64*n*(6*n^2+59*n-33) The , 2, 4, alpha coeff. is 3+59/2/n-33/2/n^2 that tends to, 3 as it should The , 2, 5, mixed moment is 0 The , 2, 5, alpha coeff. is 0 that tends to 0, as it should The , 2, 6, mixed moment is 256*n*(30*n^3+615*n^2-778*n+261) The , 2, 6, alpha coeff. is 15+615/2/n-389/n^2+261/2/n^3 that tends to, 15 as it should The , 2, 7, mixed moment is 0 The , 2, 7, alpha coeff. is 0 that tends to 0, as it should The , 3, 0, mixed moment is 0 The , 3, 0, alpha coeff. is 0 that tends to 0, as it should The , 3, 1, mixed moment is 0 The , 3, 1, alpha coeff. is 0 that tends to 0, as it should The , 3, 2, mixed moment is 64*n*(6*n+13) The , 3, 2, alpha coeff. is 3/2*8^(1/2)*(1/n)^(1/2)+13/4*8^(1/2)*(1/n)^(3/2) that tends to 0, as it should The , 3, 3, mixed moment is 0 The , 3, 3, alpha coeff. is 0 that tends to 0, as it should The , 3, 4, mixed moment is 512*n*(18*n^2+87*n-76) The , 3, 4, alpha coeff. is 9*8^(1/2)*(1/n)^(1/2)+87/2*8^(1/2)*(1/n)^(3/2)-38*8^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 3, 5, mixed moment is 0 The , 3, 5, alpha coeff. is 0 that tends to 0, as it should The , 3, 6, mixed moment is 512*n*(540*n^3+4680*n^2-8703*n+4046) The , 3, 6, alpha coeff. is 135/2*8^(1/2)*(1/n)^(1/2)+585*8^(1/2)*(1/n)^(3/2)-8703/8*8^(1/2)*(1/n)^(5/2)+ 2023/4*8^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 3, 7, mixed moment is 0 The , 3, 7, alpha coeff. is 0 that tends to 0, as it should The , 4, 0, mixed moment is 8*n*(24*n+13) The , 4, 0, alpha coeff. is 3+13/8/n that tends to, 3 as it should The , 4, 1, mixed moment is 0 The , 4, 1, alpha coeff. is 0 that tends to 0, as it should The , 4, 2, mixed moment is 32*n*(24*n^2+157*n+270) The , 4, 2, alpha coeff. is 3+157/8/n+135/4/n^2 that tends to, 3 as it should The , 4, 3, mixed moment is 0 The , 4, 3, alpha coeff. is 0 that tends to 0, as it should The , 4, 4, mixed moment is 64*n*(144*n^3+2934*n^2+9227*n-12270) The , 4, 4, alpha coeff. is 9+1467/8/n+9227/16/n^2-6135/8/n^3 that tends to, 9 as it should The , 4, 5, mixed moment is 0 The , 4, 5, alpha coeff. is 0 that tends to 0, as it should The , 4, 6, mixed moment is 256*n*(720*n^4+30270*n^3+152313*n^2-417706*n+258240) The , 4, 6, alpha coeff. is 45+15135/8/n+152313/16/n^2-208853/8/n^3+16140/n^4 that tends to, 45 as it should The , 4, 7, mixed moment is 0 The , 4, 7, alpha coeff. is 0 that tends to 0, as it should The , 5, 0, mixed moment is 0 The , 5, 0, alpha coeff. is 0 that tends to 0, as it should The , 5, 1, mixed moment is 0 The , 5, 1, alpha coeff. is 0 that tends to 0, as it should The , 5, 2, mixed moment is 128*n*(120*n^2+585*n+857) The , 5, 2, alpha coeff. is 15/2*8^(1/2)*(1/n)^(1/2)+585/16*8^(1/2)*(1/n)^(3/2)+857/16*8^(1/2)*(1/n)^(5/2) that tends to 0, as it should The , 5, 3, mixed moment is 0 The , 5, 3, alpha coeff. is 0 that tends to 0, as it should The , 5, 4, mixed moment is 1024*n*(360*n^3+3795*n^2+8196*n-16967) The , 5, 4, alpha coeff. is 45*8^(1/2)*(1/n)^(1/2)+3795/8*8^(1/2)*(1/n)^(3/2)+2049/2*8^(1/2)*(1/n)^(5/2)-\ 16967/8*8^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 5, 5, mixed moment is 0 The , 5, 5, alpha coeff. is 0 that tends to 0, as it should The , 5, 6, mixed moment is 1024*n*(10800*n^4+202050*n^3+627495*n^2-2673780*n+2147119) The , 5, 6, alpha coeff. is 675/2*8^(1/2)*(1/n)^(1/2)+101025/16*8^(1/2)*(1/n)^(3/2)+627495/32*8^(1/2)*(1/n) ^(5/2)-668445/8*8^(1/2)*(1/n)^(7/2)+2147119/32*8^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 5, 7, mixed moment is 0 The , 5, 7, alpha coeff. is 0 that tends to 0, as it should The , 6, 0, mixed moment is 64*n*(120*n^2+195*n+152) The , 6, 0, alpha coeff. is 15+195/8/n+19/n^2 that tends to, 15 as it should The , 6, 1, mixed moment is 0 The , 6, 1, alpha coeff. is 0 that tends to 0, as it should The , 6, 2, mixed moment is 256*n*(120*n^3+1275*n^2+4787*n+5916) The , 6, 2, alpha coeff. is 15+1275/8/n+4787/8/n^2+1479/2/n^3 that tends to, 15 as it should The , 6, 3, mixed moment is 0 The , 6, 3, alpha coeff. is 0 that tends to 0, as it should The , 6, 4, mixed moment is 512*n*(720*n^4+22650*n^3+162897*n^2+230683*n-810306) The , 6, 4, alpha coeff. is 45+11325/8/n+162897/16/n^2+230683/16/n^3-405153/8/n^4 that tends to, 45 as it should The , 6, 5, mixed moment is 0 The , 6, 5, alpha coeff. is 0 that tends to 0, as it should The , 6, 6, mixed moment is 2048*n*(3600*n^5+230850*n^4+2692425*n^3+4684230*n^2-36181211*n+37264692) The , 6, 6, alpha coeff. is 225+115425/8/n+2692425/16/n^2+2342115/8/n^3-36181211/16/n^4+9316173/4/n^5 that tends to, 225 as it should The , 6, 7, mixed moment is 0 The , 6, 7, alpha coeff. is 0 that tends to 0, as it should The , 7, 0, mixed moment is 0 The , 7, 0, alpha coeff. is 0 that tends to 0, as it should The , 7, 1, mixed moment is 0 The , 7, 1, alpha coeff. is 0 that tends to 0, as it should The , 7, 2, mixed moment is 512*n*(1680*n^3+13650*n^2+44037*n+45446) The , 7, 2, alpha coeff. is 105/2*8^(1/2)*(1/n)^(1/2)+6825/16*8^(1/2)*(1/n)^(3/2)+44037/32*8^(1/2)*(1/n)^(5 /2)+22723/16*8^(1/2)*(1/n)^(7/2) that tends to 0, as it should The , 7, 3, mixed moment is 0 The , 7, 3, alpha coeff. is 0 that tends to 0, as it should The , 7, 4, mixed moment is 4096*n*(5040*n^4+84630*n^3+456771*n^2+327984*n-2591402) The , 7, 4, alpha coeff. is 315*8^(1/2)*(1/n)^(1/2)+42315/8*8^(1/2)*(1/n)^(3/2)+456771/16*8^(1/2)*(1/n)^(5/ 2)+20499*8^(1/2)*(1/n)^(7/2)-1295701/8*8^(1/2)*(1/n)^(9/2) that tends to 0, as it should The , 7, 5, mixed moment is 0 The , 7, 5, alpha coeff. is 0 that tends to 0, as it should The , 7, 6, mixed moment is 4096*n*(151200*n^5+4428900*n^4+36219540*n^3+18986385*n^2-512165706*n+674894632) The , 7, 6, alpha coeff. is 4725/2*8^(1/2)*(1/n)^(1/2)+1107225/16*8^(1/2)*(1/n)^(3/2)+9054885/16*8^(1/2)*(1 /n)^(5/2)+18986385/64*8^(1/2)*(1/n)^(7/2)-256082853/32*8^(1/2)*(1/n)^(9/2)+ 84361829/8*8^(1/2)*(1/n)^(11/2) that tends to 0, as it should The , 7, 7, mixed moment is 0 The , 7, 7, alpha coeff. is 0 that tends to 0, as it should [[[1, 0, n, 0, n*(3*n-2), 0, n*(15*n^2-30*n+16), 0], [0, 0, 2*n, 0, 4*n*(3*n-4) , 0, 2*n*(45*n^2-150*n+136), 0], [n, 0, n*(n+4), 0, n*(3*n+52)*(n-2), 0, n*(15* n^3+510*n^2-2624*n+3424), 0], [0, 0, 2*n*(3*n+4), 0, 4*n*(3*n+20)*(3*n-8), 0, 2 *n*(135*n^3+1170*n^2-10632*n+19264), 0], [n*(3*n-2), 0, n*(3*n^2+22*n+16), 0, n *(9*n^3+276*n^2+148*n-3872), 0, n*(45*n^4+3120*n^3+3708*n^2-159968*n+411136), 0 ], [0, 0, 2*n*(15*n^2+30*n+16), 0, 4*n*(45*n^3+390*n^2-792*n-5824), 0, 2*n*(675 *n^4+13500*n^3-48540*n^2-538320*n+2133376), 0], [n*(15*n^2-30*n+16), 0, n*(15*n ^3+150*n^2+136*n+64), 0, n*(n+4)*(45*n^3+1860*n^2-1932*n-34976), 0, n*(225*n^5+ 23400*n^4+157980*n^3-1775040*n^2-5670464*n+43600384), 0], [0, 0, 2*n*(105*n^3+ 210*n^2+168*n+64), 0, 4*n*(315*n^4+3990*n^3-2856*n^2-99264*n-209920), 0, 2*n*( 4725*n^5+144900*n^4+2940*n^3-10471920*n^2-3595392*n+220801024), 0]], [[1, 0, 2* n, 0, 4*n*(3*n-1), 0, 8*n*(15*n^2-15*n+4), 0], [0, 0, 8*n, 0, 32*n*(3*n-2), 0, 32*n*(45*n^2-75*n+34), 0], [6*n, 0, 4*n*(3*n+10), 0, 8*n*(9*n^2+105*n-118), 0, 16*n*(45*n^3+1125*n^2-2868*n+1900), 0], [0, 0, 16*n*(9*n+17), 0, 64*n*(27*n^2+ 123*n-232), 0, 64*n*(405*n^3+3510*n^2-14019*n+12908), 0], [12*n*(9*n+5), 0, 8*n *(9*n+53)*(3*n+4), 0, 16*n*(81*n^3+1962*n^2+4197*n-14756), 0, 32*n*(405*n^4+ 20880*n^3+76743*n^2-551820*n+686048), 0], [0, 0, 32*n*(135*n^2+585*n+274), 0, 128*n*(405*n^3+4185*n^2+1992*n-29708), 0, 128*n*(6075*n^4+118800*n^3+87345*n^2-\ 2672760*n+4525516), 0], [24*n*(135*n^2+225*n-236), 0, 16*n*(405*n^3+4725*n^2+ 11082*n+1960), 0, 32*n*(1215*n^4+45360*n^3+258831*n^2-269670*n-1940632), 0, 64* n*(6075*n^5+477900*n^4+4869225*n^3-9040500*n^2-99171492*n+239148880), 0], [0, 0 , 64*n*(2835*n^3+20790*n^2+21231*n-7972), 0, 256*n*(8505*n^4+141750*n^3+403893* n^2-1415208*n-3921808), 0, 256*n*(127575*n^5+3954825*n^4+20821185*n^3-115516125 *n^2-411093996*n+1582619072), 0]], [[1, 0, 3*n, 0, 3*n*(9*n-2), 0, 3*n*(135*n^2 -90*n+16), 0], [0, 0, 12*n, 0, 24*n*(9*n-4), 0, 12*n*(405*n^2-450*n+136), 0], [ 6*n, 0, 18*n*(n+4), 0, 18*n*(9*n^2+118*n-88), 0, 18*n*(135*n^3+3690*n^2-6224*n+ 2784), 0], [6*n, 0, 6*n*(39*n+98), 0, 6*n*(675*n^2+4062*n-4744), 0, 6*n*(14985* n^3+165780*n^2-411756*n+254528), 0], [12*n*(9*n+5), 0, 36*n*(9*n^2+85*n+160), 0 , 36*n*(81*n^3+2331*n^2+8518*n-15408), 0, 36*n*(1215*n^4+71145*n^3+432414*n^2-\ 1604960*n+1318400), 0], [120*n*(3*n+5), 0, 24*n*(315*n^2+1875*n+2558), 0, 24*n* (5265*n^3+71865*n^2+161814*n-480544), 0, 24*n*(115425*n^4+2849175*n^3+9793080*n ^2-59048700*n+63081488), 0], [24*n*(135*n^2+240*n+49), 0, 216*n*(45*n^3+740*n^2 +3243*n+3212), 0, 72*n*(1215*n^4+56970*n^3+500721*n^2+582822*n-3522632), 0, 72* n*(18225*n^5+1696950*n^4+24902775*n^3+38228730*n^2-500220096*n+690234656), 0], [168*n*(135*n^2+525*n-98), 0, 24*n*(14175*n^3+146475*n^2+464016*n+335612), 0, 24*n*(229635*n^4+5162535*n^3+31540320*n^2+4698948*n-244599856), 0, 24*n*( 4975425*n^5+197953875*n^4+1929483990*n^3-367111980*n^2-39016329864*n+ 70067421632), 0]], [[1, 0, 4*n, 0, 8*n*(6*n-1), 0, 32*n*(30*n^2-15*n+2), 0], [0 , 0, 16*n, 0, 128*n*(3*n-1), 0, 128*n*(90*n^2-75*n+17), 0], [8*n, 0, 32*n*(n+3) , 0, 64*n*(6*n^2+59*n-33), 0, 256*n*(30*n^3+615*n^2-778*n+261), 0], [0, 0, 64*n *(6*n+13), 0, 512*n*(18*n^2+87*n-76), 0, 512*n*(540*n^3+4680*n^2-8703*n+4046), 0], [8*n*(24*n+13), 0, 32*n*(24*n^2+157*n+270), 0, 64*n*(144*n^3+2934*n^2+9227* n-12270), 0, 256*n*(720*n^4+30270*n^3+152313*n^2-417706*n+258240), 0], [0, 0, 128*n*(120*n^2+585*n+857), 0, 1024*n*(360*n^3+3795*n^2+8196*n-16967), 0, 1024*n *(10800*n^4+202050*n^3+627495*n^2-2673780*n+2147119), 0], [64*n*(120*n^2+195*n+ 152), 0, 256*n*(120*n^3+1275*n^2+4787*n+5916), 0, 512*n*(720*n^4+22650*n^3+ 162897*n^2+230683*n-810306), 0, 2048*n*(3600*n^5+230850*n^4+2692425*n^3+4684230 *n^2-36181211*n+37264692), 0], [0, 0, 512*n*(1680*n^3+13650*n^2+44037*n+45446), 0, 4096*n*(5040*n^4+84630*n^3+456771*n^2+327984*n-2591402), 0, 4096*n*(151200*n ^5+4428900*n^4+36219540*n^3+18986385*n^2-512165706*n+674894632), 0]]]