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A282777 Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A282777 #30 Nov 21 2024 05:13:57
%S A282777 0,1,65538,43046724,4295098372,152587890630,2821196197512,
%T A282777 33232930569608,281483566907400,1853020317992013,10000305176108940,
%U A282777 45949729863572172,184889914172333328,665416609183179854,2178019803670969104,6568408813691796120
%N A282777 Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A282777 Multiplicative because A013963 is. - _Andrew Howroyd_, Jul 25 2018
%D A282777 George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012. See p. 212.
%H A282777 Seiichi Manyama, <a href="/A282777/b282777.txt">Table of n, a(n) for n = 0..1000</a>
%F A282777 a(n) = n*A013963(n) for n > 0.
%F A282777 a(n) = (2156*A282546(n) - 4156*A282000(n) + 8000*A282547(n)/3 - 2000*A282253(n)/3)/16320.
%F A282777 Sum_{k=1..n} a(k) ~ zeta(16) * n^17 / 17. - _Amiram Eldar_, Sep 06 2023
%F A282777 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282777 Multiplicative with a(p^e) = p^e * (p^(15*e+15)-1)/(p^15-1).
%F A282777 Dirichlet g.f.: zeta(s-1)*zeta(s-16). (End)
%t A282777 Table[If[n==0, 0, n * DivisorSigma[15, n]], {n, 0, 15}] (* _Indranil Ghosh_, Mar 11 2017 *)
%o A282777 (PARI) for(n=0, 15, print1(if(n==0, 0, n * sigma(n, 15)), ", ")) \\ _Indranil Ghosh_, Mar 11 2017
%Y A282777 Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), A282597 (phi_{14, 1}), this sequence (phi_{16, 1}).
%Y A282777 Cf. A282546 (E_2*E_4^4), A282000 (E_4^3*E_6), A282547 (E_2*E_4*E_6^2), A282253 (E_6^3).
%Y A282777 Cf. A013674.
%K A282777 nonn,easy,mult
%O A282777 0,3
%A A282777 _Seiichi Manyama_, Feb 21 2017