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A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A282753 #31 Aug 02 2025 03:23:31
%S A282753 0,1,516,19692,264208,1953150,10161072,40353656,135274560,387597717,
%T A282753 1007825400,2357947812,5202783936,10604499542,20822486496,38461429800,
%U A282753 69260574976,118587876786,200000421972,322687698140,516037855200,794644193952,1216701070992
%N A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A282753 Multiplicative because A013955 is. - _Andrew Howroyd_, Jul 25 2018
%H A282753 Seiichi Manyama, <a href="/A282753/b282753.txt">Table of n, a(n) for n = 0..1000</a>
%F A282753 a(n) = n^2*A013955(n) for n > 0.
%F A282753 a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.
%F A282753 Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - _Amiram Eldar_, Sep 06 2023
%F A282753 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282753 Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).
%F A282753 Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)
%F A282753 G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - _Vaclav Kotesovec_, Aug 02 2025
%t A282753 Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* _Indranil Ghosh_, Mar 12 2017 *)
%t A282753 nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A282753 (PARI) for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ _Indranil Ghosh_, Mar 12 2017
%Y A282753 Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).
%Y A282753 Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
%Y A282753 Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).
%Y A282753 Cf. A013666.
%K A282753 nonn,easy,mult
%O A282753 0,3
%A A282753 _Seiichi Manyama_, Feb 21 2017