A036039 tabf array: M_2 numbers of Abramowitz and Stegun p.831. 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 2 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 8 3 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 24 30 20 20 15 10 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 120 144 90 40 90 120 15 40 45 15 1 0 0 0 0 0 0 0 0 0 0 0 7 720 840 504 420 504 630 280 210 210 420 105 70 105 21 1 0 0 0 0 0 0 0 8 5040 5760 3360 2688 1260 3360 4032 3360 1260 1120 1344 2520 1120 1680 105 420 1120 420 112 210 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: n=9: [40320, 45360, 25920, 20160, 18144, 25920, 30240, 24192, 11340, 9072, 15120, 2240, 10080, 18144, 15120, 11340, 10080, 2520, 3024, 7560, 3360, 7560, 945, 756, 2520, 1260, 168, 378, 36, 1] n=10: [362880, 403200, 226800, 172800, 151200, 72576, 226800, 259200, 201600, 181440, 75600, 120960, 56700, 50400, 86400, 151200, 120960, 56700, 90720, 151200, 22400, 18900, 25200, 25200, 60480, 50400, 56700, 50400, 25200, 945, 6048, 18900, 8400, 25200, 4725, 1260, 5040, 3150, 240, 630, 45, 1] The row sums give A000142 (factorials): [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,...] The row polynomials are (A-St order): n=1: x[1] n=2: x[2] + x[1]^2 n=3: 2*x[3]+3*x[1]*x[2]+x[1]^3 n=4: 6*x[4]+8*x[1]*x[3]+3*x[2]^2+6*x[1]^2*x[2]+x[1]^4 n=5: 24*x[5]+30*x[1]*x[4]+20*x[2]*x[3]+20*x[1]^2*x[3]+15*x[1]*x[2]^2+10*x[1]^3*x[2]+x[1]^5 n=6: 120*x[6]+144*x[1]*x[5]+90*x[2]*x[4]+40*x[3]^2+90*x[1]^2*x[4]+120*x[1]*x[2]*x[3]+15*x[2]^3 + +40*x[1]^3*x[3]+45*x[1]^2*x[2]^2+15*x[1]^4*x[2]+x[1]^6 n=7: 720*x[7]+840*x[1]*x[6]+504*x[2]*x[5]+420*x[3]*x[4]+504*x[1]^2*x[5]+630*x[1]*x[2]*x[4]+ +280*x[1]*x[3]^2+210*x[2]^2*x[3]+210*x[1]^3*x[4]+420*x[1]^2*x[2]*x[3]+105*x[1]*x[2]^3+ +70*x[1]^4*x[3]+105*x[1]^3*x[2]^2+21*x[1]^5*x[2]+x[1]^7 n=8: 5040*x[8]+5760*x[1]*x[7]+3360*x[2]*x[6]+2688*x[3]*x[5]+1260*x[4]^2+3360*x[1]^2*x[6]+ + 4032*x[1]*x[2]*x[5]+3360*x[1]*x[3]*x[4]+1260*x[2]^2*x[4]+1120*x[2]*x[3]^2+ + 1344*x[1]^3*x[5]+2520*x[1]^2*x[2]*x[4]+1120*x[1]^2*x[3]^2+1680*x[1]*x[2]^2*x[3]+ +105*x[2]^4+420*x[1]^4*x[4]+1120*x[1]^3*x[2]*x[3]+420*x[1]^2*x[2]^3+112*x[1]^5*x[3]+ +210*x[1]^4*x[2]^2+28*x[1]^6*x[2]+x[1]^8 n=9: 40320*x[9] + 45360*x[1]*x[8] + 25920*x[2]*x[7] + 20160*x[3]*x[6] + 18144*x[4]*x[5]+ + 25920*x[1]^2*x[7] + 30240*x[1]*x[2]*x[6] + 24192*x[1]*x[3]*x[5] + 11340*x[1]*x[4]^2 + + 9072*x[2]^2*x[5] + 15120*x[2]*x[3]*x[4] + 2240*x[3]^3 + 10080*x[1]^3*x[6] + 18144*x[1]^2*x[2]*x[5]+ + 15120*x[1]^2*x[3]*x[4] + 11340*x[1]*x[2]^2*x[4] + 10080*x[1]*x[2]*x[3]^2 + 2520*x[2]^3*x[3] + + 3024*x[1]^4*x[5] + 7560*x[1]^3*x[2]*x[4] + 3360*x[1]^3*x[3]^2 + 7560*x[1]^2*x[2]^2*x[3] + + 945*x[1]*x[2]^4 + 756*x[1]^5*x[4] + 2520*x[1]^4*x[2]*x[3] + 1260*x[1]^3*x[2]^3 + 168*x[1]^6*x[3] + + 378*x[1]^5*x[2]^2 + 36*x[1]^7*x[2] + x[1]^9 n=10: 362880*x[10] + 403200*x[1]*x[9] + 226800*x[2]*x[8] + 172800*x[3]*x[7] + 151200*x[4]*x[6] + + 72576*x[5]^2 + 226800*x[1]^2*x[8] + 259200*x[1]*x[2]*x[7] + 201600*x[1]*x[3]*x[6] + + 181440*x[1]*x[4]*x[5] + 75600*x[2]^2*x[6] + 120960*x[2]*x[3]*x[5] + 56700*x[2]*x[4]^2 + + 50400*x[3]^2*x[4] + 86400*x[1]^3*x[7] + 151200*x[1]^2*x[2]*x[6] + 120960*x[1]^2*x[3]*x[5] + + 56700*x[1]^2*x[4]^2 + 90720*x[1]*x[2]^2*x[5]+ 151200*x[1]*x[2]*x[3]*x[4] + 22400*x[1]*x[3]^3 + + 18900*x[2]^3*x[4] + 25200*x[2]^2*x[3]^2 + 25200*x[1]^4*x[6] + 60480*x[1]^3*x[2]*x[5] + + 50400*x[1]^3*x[3]*x[4] + 56700*x[1]^2*x[2]^2*x[4] + 50400*x[1]^2*x[2]*x[3]^2 + + 25200*x[1]*x[2]^3*x[3] + 945*x[2]^5 + 6048*x[1]^5*x[5] + 18900*x[1]^4*x[2]*x[4]+ + 8400*x[1]^4*x[3]^2 + 25200*x[1]^3*x[2]^2*x[3] + 4725*x[1]^2*x[2]^4 + 1260*x[1]^6*x[4]+ + 5040*x[1]^5*x[2]*x[3] + 3150*x[1]^4*x[2]^3 + 240*x[1]^7*x[3] + 630*x[1]^6*x[2]^2+ + 45*x[1]^8*x[2] + x[1]^10 ############################################## e.o.f. ############################################################