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A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A282781 #31 Aug 02 2025 07:31:40
%S A282781 0,1,264,6588,67648,390750,1739232,5765144,17318400,43224597,
%T A282781 103158000,214360212,445665024,815732918,1521998016,2574261000,
%U A282781 4433514496,6975762354,11411293608,16983569900,26433456000,37980768672,56591095968,78310997448
%N A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A282781 Multiplicative because A001160 is. - _Andrew Howroyd_, Jul 25 2018
%H A282781 Seiichi Manyama, <a href="/A282781/b282781.txt">Table of n, a(n) for n = 0..1000</a>
%F A282781 a(n) = n^3*A001160(n) for n > 0.
%F A282781 a(n) = (6*A282752(n) - 2*A282780(n) - 6*A282102(n) + A008411(n) + A280869(n))/5184.
%F A282781 Sum_{k=1..n} a(k) ~ zeta(6) * n^9 / 9. - _Amiram Eldar_, Sep 06 2023
%F A282781 From _Amiram Eldar_, Oct 31 2023: (Start)
%F A282781 Multiplicative with a(p^e) = p^(3*e) * (p^(5*e+5)-1)/(p^5-1).
%F A282781 Dirichlet g.f.: zeta(s-3)*zeta(s-8). (End)
%F A282781 G.f.: Sum_{k>=1} k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - _Vaclav Kotesovec_, Aug 02 2025
%t A282781 a[0]=0;a[n_]:=(n^3)*DivisorSigma[5,n];Table[a[n],{n,0,23}] (* _Indranil Ghosh_, Feb 21 2017 *)
%t A282781 nmax = 30; CoefficientList[Series[Sum[k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A282781 (PARI) a(n) = if (n==0, 0, n^3*sigma(n, 5)); \\ _Michel Marcus_, Feb 21 2017
%Y A282781 Cf. A282211 (phi_{4, 3}), A282213 (phi_{6, 3}), this sequence (phi_{8, 3}).
%Y A282781 Cf. A282752 (E_2^2*E_4^2), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
%Y A282781 Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), A282751 (n^2*sigma_5(n)), this sequence (n^3*sigma_5(n)).
%Y A282781 Cf. A013664.
%K A282781 nonn,easy,mult
%O A282781 0,3
%A A282781 _Seiichi Manyama_, Feb 21 2017