A036040 tabf array: M_3 numbers of Abramowitz and Stegun. Partitions of n listed in Abramowitz-Stegun order p. 831-2 (see the main page for the reference). n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... 1 1 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 3 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 4 3 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 5 10 10 15 10 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 6 15 10 15 60 15 20 45 15 1 0 0 0 0 0 0 0 0 0 0 0 7 1 7 21 35 21 105 70 105 35 210 105 35 105 21 1 0 0 0 0 0 0 0 8 1 8 28 56 35 28 168 280 210 280 56 420 280 840 105 70 560 420 56 210 28 1 . . . n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... The next two rows, for n=9 and 10 are: [1, 9, 36, 84, 126, 36, 252, 504, 315, 378, 1260, 280, 84, 756, 1260, 1890, 2520, 1260, 126, 1260, 840, 3780, 945, 126, 1260, 1260, 84, 378, 36, 1] [1, 10, 45, 120, 210, 126, 45, 360, 840, 1260, 630, 2520, 1575, 2100, 120, 1260, 2520, 1575, 3780, 12600, 2800, 3150, 6300, 210, 2520, 4200, 9450, 12600, 12600, 945, 252, 3150, 2100, 12600, 4725, 210, 2520, 3150, 120, 630, 45, 1] ######################################################################################################## A(t) = exp(x[k]*(t^k)/k! ,k=1..infinity) is the exponential generating function (e.g.f.) of the n-variate polynomials P3_n(x[1],...,x[n]) with p(n):=A000041(n)(partition numbers) terms. These polynomials are, for n=1,...10: n=1: 1*x[1] n=2: 1*x[2] +1*x[1]^2 n=3: 1*x[3] +3*x[1]*x[2] +1*x[1]^3 n=4: 1*x[4] +4*x[1]*x[3] +3*x[2]^2 +6*x[1]^2*x[2] +1*x[1]^4 n=5: 1*x[5] +5*x[1]*x[4] +10*x[2]*x[3] +10*x[1]^2*x[3] +15*x[1]*x[2]^2 +10*x[1]^3*x[2] +1*x[1]^5 n=6: 1*x[6] +6*x[1]*x[5] +15*x[2]*x[4] +10*x[3]^2 +15*x[1]^2*x[4] +60*x[1]*x[2]*x[3] +15*x[2]^3 +20*x[1]^3*x[3] +45*x[1]^2*x[2]^2 +15*x[1]^4*x[4] +1*x[1]^6 n=7: 1*x[7] +7*x[1]*x[6] +21*x[2]*x[5] +35*x[3]*x[4] +21*x[1]^2*x[5] +105*x[1]*x[2]*x[4] +70*x[1]*x[3]^2 +105*x[2]^2*x[3] +35*x[1]^3*x[4] +210*x[1]^2*x[2]*x[3] +105*x[1]*x[2]^3 +35*x[1]^4*x[3] +105*x[1]^3*x[2]^2 +21*x[1]^5*x[2] +1*x[1]^7 n=8: 1*x[8] +8*x[1]*x[7] +28*x[2]*x[6] +56*x[3]*x[5] +35*x[4]^2 +28*x[1]^2*x[6] +168*x[1]*x[2]*x[5] +280*x[1]*x[3]*x[4] +210*x[2]^2*x[4] +280*x[2]*x[3]^2] +56*x[1]^3*x[5] +420*x[1]^2*x[2]*x[4] +280*x[1]^2*x[3]^2 +840*x[1]*x[2]^2*x[3] +105*x[2]^4 +70*x[1]^4*x[4] +560*x[1]^3*x[2]*x[3] +420*x[1]^2*x[2]^3 +56*]^5*x[3] +210*x[1]^4*x[2]^2 +28*x[1]^6*x[2] +1*x[1]^8 n=9: 1*x[9] +9*x[1]*x[8] +36*x[2]*x[7] +84*x[3]*x[6] +126*x[4]*x[5] +36*x[1]^2*x[7] +252*x[1]*x[2]*x[6] +504*x[1]*x[3]*x[5] +315*x[1]*x[4]^2 +378*x[2]^2*x[5] +1260*x[2]*x[3]*x[4] +280*x[3]^3 +84*x[1]^3*x[6] +756*x[1]^2*x[2]*x[5] +1260*x[1]^2*x[3]*x[4] +1890*x[1]*x[2]^2*x[4] +2520*x[1]*x[2]*x[3]^2+1260*x[2]^3*x[3] +126*x[1]^4*x[5] +1260*x[1]^3*x[2]*x[4] +840*x[1]^3*x[3]^2 +3780*x[1]^2*x[2]^2*x[3] +945*x[1]*x[2]^4 +126*x[1]^5*x[4] +1260*x[1]^4*x[2]*x[3] +1260*x[1]^3*x[2]^3] +84*x[1]^6*x[3] +378*x[1]^5*x[2]^2 +36*x[1]^7*x[2] +1*x[1]^9 n=10: 1*x[10] +10*x[1]*x[9] +45*x[2]*x[8] +120*x[3]*x[7] +210*x[4]*x[6] +126*x[5]^2 +45*x[1]^2*x[8] +360*x[1]*x[2]*x[7] +840*x[1]*x[3]*x[6] +1260*x[1]*x[4]*x[5] +630*x[2]^2*x[6] +2520*x[2]*x[3]*x[5] +1575*x[2]*x[4]^2 +2100*x[3]^2*x[4] +120*x[1]^3*x[7] +1260*x[1]^2*x[2]*x[6] +2520*x[1]^2*x[3]*x[5] +1575*x[1]^2*x[4]^2 +3780*x[1]*x[2]^2*x[5] +12600*x[1]*x[2]*x[3]*x[4] +2800*x[1]*x[3]^3 +3150*x[2]^3*x[4] +6300*x[2]^2*x[3]^2 +210*x[1]^4*x[6] +2520*x[1]^3*x[2]*x[5] +4200*x[1]^3*x[3]*x[4] +9450*x[1]^2*x[2]^2*x[4] +12600*x[1]^2*x[2]*x[3]^2 +12600*x[1]*x[2]^3*x[3] +945*x[2]^5 +252*x[1]^5*x[5] +3150*x[1]^4*x[2]*x[4] +2100*x[1]^4*x[3]^2 +12600*x[1]^3*x[2]^2*x[3] +4725*x[1]^2*x[2]^4 +210*x[1]^6*x[4] +2520*x[1]^5*x[2]*x[3] +3150*x[1]^4*x[2]^3 +120*x[1]^7*x[3] +630*x[1]^6*x[2]^2 +45*x[1]^8*x[2] +1*x[1]^10 #######################################################################################################