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%I #7 Nov 04 2016 09:53:32
%S 1,511,9841,174251,488281,6017605,6725601,50955971,72636421,276964061,
%T 235794769,2234070293,883708281,3698977205,5148057541,13910980083,
%U 7411742281,46982039533,17927094321,99343345101,69493620405
%N Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=8.
%F 1/8!*(sigma[1](n)^8 + 28*sigma[1](n)^6*sigma[2](n) + 112*sigma[1](n)^5*sigma[3](n) + 210*sigma[1](n)^4*sigma[2](n)^2 + 420*sigma[1](n)^4*sigma[4](n) + 1120*sigma[1](n)^3*sigma[2](n)*sigma[3](n) + 420*sigma[1](n)^2*sigma[2](n)^3 + 1344*sigma[1](n)^3*sigma[5](n) + 2520*sigma[1](n)^2*sigma[2](n)*sigma[4](n) + 1120*sigma[1](n)^2*sigma[3](n)^2 + 1680*sigma[1](n)*sigma[2](n)^2*sigma[3](n) + 105*sigma[2](n)^4 + 3360*sigma[1](n)^2*sigma[6](n) + 4032*sigma[1](n)*sigma[2](n)*sigma[5](n) + 3360*sigma[1](n)*sigma[3](n)*sigma[4](n) + 1260*sigma[2](n)^2*sigma[4](n) + 1120*sigma[2](n)*sigma[3](n)^2 + 5760*sigma[7](n)*sigma[1](n) + 3360*sigma[2](n)*sigma[6](n) + 2688*sigma[3](n)*sigma[5](n) + 1260*sigma[4](n)^2 + 5040*sigma[8](n)).
%t CIP8 = CycleIndexPolynomial[SymmetricGroup[8], Array[x, 8]]; a[n_] := CIP8 /. x[k_] -> DivisorSigma[k, n]; Array[a, 21] (* _Jean-François Alcover_, Nov 04 2016 *)
%Y Cf. A067692, A068020-A068024, A068026-A068027, A000203, A001157-A001160, A013954-A013956.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Feb 08 2002