cp's OEIS Frontend

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A280021 #38 Oct 30 2023 01:44:45
%S A280021 0,1,2052,177156,4202512,48828150,363524112,1977326792,8606744640,
%T A280021 31382654013,100195363800,285311670732,744500215872,1792160394206,
%U A280021 4057474577184,8650199741400,17626613022976,34271896307922,64397206034676,116490258898580,205200886312800
%N A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A280021 Multiplicative because A013957 is. - _Andrew Howroyd_, Jul 23 2018
%H A280021 Seiichi Manyama, <a href="/A280021/b280021.txt">Table of n, a(n) for n = 0..1000</a>
%F A280021 a(n) = n^2*A013957(n) for n > 0.
%F A280021 a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
%F A280021 Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - _Amiram Eldar_, Sep 06 2023
%F A280021 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A280021 Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
%F A280021 Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)
%t A280021 Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* _Indranil Ghosh_, Mar 12 2017 *)
%o A280021 (PARI) for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ _Indranil Ghosh_, Mar 12 2017
%Y A280021 Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
%Y A280021 Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
%Y A280021 Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
%Y A280021 Cf. A013668 (zeta(10)).
%K A280021 nonn,easy,mult
%O A280021 0,3
%A A280021 _Seiichi Manyama_, Feb 22 2017