More on tabf A102189(n,k) the row reversed A036039 tabf. The row lengths are given by the partition sequence [1,2,3,5,7,11,15,...]=A000041(n),n>=1. a(n,k) tabf head (staircase) for A102189 n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 3 2 0 0 0 0 0 0 0 0 0 0 0 0 4 1 6 3 8 6 0 0 0 0 0 0 0 0 0 0 5 1 10 15 20 20 30 24 0 0 0 0 0 0 0 0 6 1 15 45 40 15 120 90 40 90 144 120 0 0 0 0 7 1 21 105 70 105 420 210 210 280 630 504 420 504 840 720 . . . The next rows, n=9..12, are: n=8: [1, 28, 210, 112, 420, 1120, 420, 105, 1680, 1120, 2520, 1344, 1120, 1260, 3360, 4032, 3360, 1260, 2688, 3360, 5760, 5040] n=9: [1, 36, 378, 168, 1260, 2520, 756, 945, 7560, 3360, 7560, 3024, 2520, 10080, 11340, 15120, 18144, 10080, 2240, 15120, 9072, 11340, 24192, 30240, 25920, 18144, 20160, 25920, 45360, 40320] n=10: [1, 45, 630, 240, 3150, 5040, 1260, 4725, 25200, 8400, 18900, 6048, 945, 25200, 50400, 56700, 50400, 60480, 25200, 25200, 18900, 22400, 151200, 90720, 56700, 120960, 151200, 86400, 50400, 56700, 120960, 75600, 181440, 201600, 259200, 226800, 72576, 151200, 172800, 226800, 403200, 362880] n=11: [1, 55, 990, 330, 6930, 9240, 1980, 17325, 69300, 18480, 41580, 11088, 10395, 138600, 184800, 207900, 138600, 166320, 55440, 34650, 277200, 207900, 123200, 831600, 498960, 207900, 443520, 554400, 237600, 123200, 415800, 623700, 166320, 554400, 1330560, 831600, 997920, 1108800, 1425600, 831600, 415800, 443520, 997920, 1108800, 712800, 798336, 1663200, 1900800, 2494800, 2217600, 1330560, 1425600, 1663200, 2217600, 3991680, 3628800] n=12: [1, 66, 1485, 440, 13860, 15840, 2970, 51975, 166320, 36960, 83160, 19008, 62370, 554400, 554400, 623700, 332640, 399168, 110880, 10395, 415800, 1663200, 1247400, 492800, 3326400, 1995840, 623700, 1330560, 1663200, 570240, 554400, 1478400, 311850, 4989600, 3326400, 3742200, 1995840, 7983360, 4989600, 3991680, 4435200, 5702400, 2494800, 246400, 3326400, 1871100, 3991680, 4989600, 5322240, 11975040, 1663200, 13305600, 8553600, 4790016, 9979200, 11404800, 14968800, 8870400, 1247400, 7983360, 4435200, 4790016, 9979200, 11404800, 7484400, 15966720, 17107200, 19958400, 26611200, 23950080, 6652800, 13685760, 14968800, 17740800, 23950080, 43545600, 39916800] ############################################################################################################# These M_2 multinomial coefficients (Abramowitz-Stegun p. 831-2) appear as coefficients of the n-variate cycle index polynomials for the symmetric group S_n, called Z(S_n,x). The monomials for these polynomials are (in the above given order of coefficients *1/n!), for n=1..10: n=1: [x[1]] n=2: [1/2*x[1]^2, 1/2*x[2]] n=3: [1/6*x[1]^3, 1/2*x[1]*x[2], 1/3*x[3]] n=4: [1/24*x[1]^4, 1/4*x[1]^2*x[2], 1/8*x[2]^2, 1/3*x[1]*x[3], 1/4*x[4]] n=5: [1/120*x[1]^5, 1/12*x[1]^3*x[2], 1/8*x[1]*x[2]^2, 1/6*x[1]^2*x[3], 1/6*x[2]*x[3], 1/4*x[1]*x[4], 1/5*x[5]] n=6: [1/720*x[1]^6, 1/48*x[1]^4*x[2], 1/16*x[1]^2*x[2]^2, 1/18*x[1]^3*x[3], 1/48*x[2]^3, 1/6*x[1]*x[2]*x[3], 1/8*x[1]^2*x[4], 1/18*x[3]^2, 1/8*x[2]*x[4], 1/5*x[1]*x[5], 1/6*x[6]] n=7: [1/5040*x[1]^7, 1/240*x[1]^5*x[2], 1/48*x[1]^3*x[2]^2, 1/72*x[1]^4*x[3], 1/48*x[1]*x[2]^3, 1/12*x[1]^2*x[2]*x[3], 1/24*x[1]^3*x[4], 1/24*x[2]^2*x[3], 1/18*x[1]*x[3]^2, 1/8*x[1]*x[2]*x[4], 1/10*x[1]^2*x[5], 1/12*x[3]*x[4], 1/10*x[2]*x[5], 1/6*x[1]*x[6], 1/7*x[7]] n=8: [1/40320*x[1]^8, 1/1440*x[1]^6*x[2], 1/192*x[1]^4*x[2]^2, 1/360*x[1]^5*x[3], 1/96*x[1]^2*x[2]^3, 1/36*x[1]^3*x[2]*x[3], 1/96*x[1]^4*x[4], 1/384*x[2]^4, 1/24*x[1]*x[2]^2*x[3], 1/36*x[1]^2*x[3]^2, 1/16*x[1]^2*x[2]*x[4], 1/30*x[1]^3*x[5], 1/36*x[2]*x[3]^2, 1/32*x[2]^2*x[4], 1/12*x[1]*x[3]*x[4], 1/10*x[1]*x[2]*x[5], 1/12*x[1]^2*x[6], 1/32*x[4]^2, 1/15*x[3]*x[5], 1/12*x[2]*x[6], 1/7*x[1]*x[7], 1/8*x[8]] n=9: [1/362880*x[1]^9, 1/10080*x[1]^7*x[2], 1/960*x[1]^5*x[2]^2, 1/2160*x[1]^6*x[3], 1/288*x[1]^3*x[2]^3, 1/144*x[1]^4*x[2]*x[3], 1/480*x[1]^5*x[4], 1/384*x[1]*x[2]^4, 1/48*x[1]^2*x[2]^2*x[3], 1/108*x[1]^3*x[3]^2, 1/48*x[1]^3*x[2]*x[4], 1/120*x[1]^4*x[5], 1/144*x[2]^3*x[3], 1/36*x[1]*x[2]*x[3]^2, 1/32*x[1]*x[2]^2*x[4], 1/24*x[1]^2*x[3]*x[4], 1/20*x[1]^2*x[2]*x[5], 1/36*x[1]^3*x[6], 1/162*x[3]^3, 1/24*x[2]*x[3]*x[4], 1/40*x[2]^2*x[5], 1/32*x[1]*x[4]^2, 1/15*x[1]*x[3]*x[5], 1/12*x[1]*x[2]*x[6], 1/14*x[1]^2*x[7], 1/20*x[4]*x[5], 1/18*x[3]*x[6], 1/14*x[2]*x[7], 1/8*x[1]*x[8], 1/9*x[9]] n=10: [1/362880*x[1]^9, 1/10080*x[1]^7*x[2], 1/960*x[1]^5*x[2]^2, 1/2160*x[1]^6*x[3], 1/288*x[1]^3*x[2]^3, 1/144*x[1]^4*x[2]*x[3], 1/480*x[1]^5*x[4], 1/384*x[1]*x[2]^4, 1/48*x[1]^2*x[2]^2*x[3], 1/108*x[1]^3*x[3]^2, 1/48*x[1]^3*x[2]*x[4], 1/120*x[1]^4*x[5], 1/144*x[2]^3*x[3], 1/36*x[1]*x[2]*x[3]^2, 1/32*x[1]*x[2]^2*x[4], 1/24*x[1]^2*x[3]*x[4], 1/20*x[1]^2*x[2]*x[5], 1/36*x[1]^3*x[6], 1/162*x[3]^3, 1/24*x[2]*x[3]*x[4], 1/40*x[2]^2*x[5], 1/32*x[1]*x[4]^2, 1/15*x[1]*x[3]*x[5], 1/12*x[1]*x[2]*x[6], 1/14*x[1]^2*x[7], 1/20*x[4]*x[5], 1/18*x[3]*x[6], 1/14*x[2]*x[7], 1/8*x[1]*x[8], 1/9*x[9]] ####################################################################################################################