More comments on A102190: Cycle index polynomials for the cyclic roup C_n. The cycle index polynomials are, for n=1..20: n Z(C_n,x) 1 x[1] 2 1/2*x[1]^2+1/2*x[2] 3 1/3*x[1]^3+2/3*x[3] 4 1/4*x[1]^4+1/4*x[2]^2+1/2*x[4] 5 1/5*x[1]^5+4/5*x[5] 6 1/6*x[1]^6+1/6*x[2]^3+1/3*x[3]^2+1/3*x[6] 7 1/7*x[1]^7+6/7*x[7] 8 1/8*x[1]^8+1/8*x[2]^4+1/4*x[4]^2+1/2*x[8] 9 1/9*x[1]^9+2/9*x[3]^3+2/3*x[9] 10 1/10*x[1]^10+1/10*x[2]^5+2/5*x[5]^2+2/5*x[10] 11 1/11*x[1]^11+10/11*x[11] 12 1/12*x[1]^12+1/12*x[2]^6+1/6*x[3]^4+1/6*x[4]^3+1/6*x[6]^2+1/3*x[12] 13 1/13*x[1]^13+12/13*x[13] 14 1/14*x[1]^14+1/14*x[2]^7+3/7*x[7]^2+3/7*x[14] 15 1/15*x[1]^15+2/15*x[3]^5+4/15*x[5]^3+8/15*x[15] 16 1/16*x[1]^16+1/16*x[2]^8+1/8*x[4]^4+1/4*x[8]^2+1/2*x[16] 17 1/17*x[1]^17+16/17*x[17] 18 1/18*x[1]^18+1/18*x[2]^9+1/9*x[3]^6+1/9*x[6]^3+1/3*x[9]^2+1/3*x[18] 19 1/19*x[1]^19+18/19*x[19] 20 1/20*x[1]^20+1/20*x[2]^10+1/10*x[4]^5+1/5*x[5]^4+1/5*x[10]^2+2/5*x[20]] ----------------------------------------------------------- The corresponding list of lists of coefficients of n*Z(C_n,x), n=1..20, is: [[1], [1, 1], [1, 2], [1, 1, 2], [1, 4], [1, 1, 2, 2], [1, 6], [1, 1, 2, 4], [1, 2, 6], [1, 1, 4, 4], [1, 10], [1, 1, 2, 2, 2, 4], [1, 12], [1, 1, 6, 6], [1, 2, 4, 8], [1, 1, 2, 4, 8], [1, 16], [1, 1, 2, 2, 6, 6], [1, 18], [1, 1, 2, 4, 4, 8]] Or, as sequence: [1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 2, 1, 6, 1, 1, 2, 4, 1, 2, 6, 1, 1, 4, 4, 1, 10, 1, 1, 2, 2, 2, 4, 1, 12, 1, 1, 6, 6, 1, 2, 4, 8, 1, 1, 2, 4, 8, 1, 16, 1, 1, 2, 2, 6, 6, 1, 18, 1, 1, 2, 4, 4, 8] ------------------------------------------------------------ The list of lists for n=1..50 is: [[1], [1, 1], [1, 2], [1, 1, 2], [1, 4], [1, 1, 2, 2], [1, 6], [1, 1, 2, 4], [1, 2, 6], [1, 1, 4, 4], [1, 10], [1, 1, 2, 2, 2, 4], [1, 12], [1, 1, 6, 6], [1, 2, 4, 8], [1, 1, 2, 4, 8], [1, 16], [1, 1, 2, 2, 6, 6], [1, 18], [1, 1, 2, 4, 4, 8], [1, 2, 6, 12], [1, 1, 10, 10], [1, 22], [1, 1, 2, 2, 2, 4, 4, 8], [1, 4, 20], [1, 1, 12, 12], [1, 2, 6, 18], [1, 1, 2, 6, 6, 12], [1, 28], [1, 1, 2, 4, 2, 4, 8, 8], [1, 30], [1, 1, 2, 4, 8, 16], [1, 2, 10, 20], [1, 1, 16, 16], [1, 4, 6, 24], [1, 1, 2, 2, 2, 6, 4, 6, 12], [1, 36], [1, 1, 18, 18], [1, 2, 12, 24], [1, 1, 2, 4, 4, 4, 8, 16], [1, 40], [1, 1, 2, 2, 6, 6, 12, 12], [1, 42], [1, 1, 2, 10, 10, 20], [1, 2, 4, 6, 8, 24], [1, 1, 22, 22], [1, 46], [1, 1, 2, 2, 2, 4, 4, 8, 8, 16], [1, 6, 42], [1, 1, 4, 4, 20, 20]] ------------------------------------------------------------ The coefficients in row n come from phi(k)=A000010(k), the divisors k of n, when taken in increasing order. Euler's totient function phi(k) is for k=1..50: [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20], and the list of divisors for n=1..50 is, in increasing order: [[1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6], [1, 7], [1, 2, 4, 8], [1, 3, 9], [1, 2, 5, 10], [1, 11], [1, 2, 3, 4, 6, 12], [1, 13], [1, 2, 7, 14], [1, 3, 5, 15], [1, 2, 4, 8, 16], [1, 17], [1, 2, 3, 6, 9, 18], [1, 19], [1, 2, 4, 5, 10, 20], [1, 3, 7, 21], [1, 2, 11, 22], [1, 23], [1, 2, 3, 4, 6, 8, 12, 24], [1, 5, 25], [1, 2, 13, 26], [1, 3, 9, 27], [1, 2, 4, 7, 14, 28], [1, 29], [1, 2, 3, 5, 6, 10, 15, 30], [1, 31], [1, 2, 4, 8, 16, 32], [1, 3, 11, 33], [1, 2, 17, 34], [1, 5, 7, 35], [1, 2, 3, 4, 6, 9, 12, 18, 36], [1, 37], [1, 2, 19, 38], [1, 3, 13, 39], [1, 2, 4, 5, 8, 10, 20, 40], [1, 41], [1, 2, 3, 6, 7, 14, 21, 42], [1, 43], [1, 2, 4, 11, 22, 44], [1, 3, 5, 9, 15, 45], [1, 2, 23, 46], [1, 47], [1, 2, 3, 4, 6, 8, 12, 16, 24, 48], [1, 7, 49], [1, 2, 5, 10, 25, 50]] ################################################### eof ############################################################