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A386785 a(n) = n^4*sigma_5(n).

%I A386785 #20 Aug 04 2025 04:11:24
%S A386785 0,1,528,19764,270592,1953750,10435392,40356008,138547200,389021373,
%T A386785 1031580000,2357962332,5347980288,10604527934,21307972224,38613915000,
%U A386785 70936231936,118587960018,205403284944,322687828100,528669120000,797596142112,1245004111296,1801152941304,2738246860800
%N A386785 a(n) = n^4*sigma_5(n).
%H A386785 Vincenzo Librandi, <a href="/A386785/b386785.txt">Table of n, a(n) for n = 0..10000</a>
%F A386785 G.f.: Sum_{k>=1} k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10.
%F A386785 a(n) = (4*A386813(n) + 2*A282549(n) - A386814(n) - 6*A282792(n) - A058550(n) + 2*A282576(n))/3456.
%F A386785 a(n) = n^4*A001160(n).
%F A386785 Dirichlet g.f.: zeta(s-4)*zeta(s-9). - _R. J. Mathar_, Aug 03 2025
%t A386785 Table[n^4*DivisorSigma[5, n], {n, 0, 30}]
%t A386785 (* or *)
%t A386785 nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10, {k, 1, nmax}], {x, 0, nmax}], x]
%t A386785 (* or *)
%t A386785 terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(4*E2[x]^3*E4[x]^2 + 2*E2[x]*E4[x]^3 - E2[x]^4*E6[x] - 6*E2[x]^2*E4[x]*E6[x] - E4[x]^2*E6[x] + 2*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]
%o A386785 (Magma) [0] cat [n^4*DivisorSigma(5, n): n in [1..35]]; // _Vincenzo Librandi_, Aug 04 2025
%Y A386785 Cf. A280022, A386783, A280025, A386784, A386786, A386787, A386788.
%Y A386785 Cf. A001160, A282050, A282751, A282781.
%Y A386785 Cf. A386813, A282549, A386814, A282792, A058550, A282576.
%K A386785 nonn,mult
%O A386785 0,3
%A A386785 _Vaclav Kotesovec_, Aug 02 2025