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A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

%I A282060 #45 Aug 01 2025 10:06:04
%S A282060 0,1,258,6564,66052,390630,1693512,5764808,16909320,43066413,
%T A282060 100782540,214358892,433565328,815730734,1487320464,2564095320,
%U A282060 4328785936,6975757458,11111134554,16983563060,25801892760,37840199712,55304594136,78310985304,110992776480
%N A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C A282060 Multiplicative because A013955 is. - _Andrew Howroyd_, Jul 25 2018
%H A282060 Seiichi Manyama, <a href="/A282060/b282060.txt">Table of n, a(n) for n = 0..1000</a>
%F A282060 G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F A282060 a(n) = (A282101(n) - A013974(n))/720. - _Seiichi Manyama_, Feb 10 2017
%F A282060 If p is a prime, a(p) = p^8 + p = A196288(p). - _Seiichi Manyama_, Feb 10 2017
%F A282060 a(n) = n*A013955(n) for n > 0. - _Seiichi Manyama_, Feb 18 2017
%F A282060 Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - _Amiram Eldar_, Sep 06 2023
%F A282060 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282060 Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
%F A282060 Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
%F A282060 G.f. Sum_{k>=1} k^8*x^(k-1)/(1 - x^k)^2. - _Vaclav Kotesovec_, Aug 01 2025
%e A282060 a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.
%t A282060 terms = 25;
%t A282060 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A282060 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A282060 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A282060 E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%t A282060 Table[n * DivisorSigma[7, n], {n, 0, 24}] (* _Amiram Eldar_, Sep 06 2023 *)
%t A282060 nmax = 40; CoefficientList[Series[x*Sum[k^8*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 01 2025 *)
%o A282060 (PARI) a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ _Andrew Howroyd_, Jul 25 2018
%Y A282060 Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), this sequence (phi_{8, 1}).
%Y A282060 Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
%Y A282060 Cf. A013955, A013666, A196288.
%K A282060 nonn,easy,mult
%O A282060 0,3
%A A282060 _Seiichi Manyama_, Feb 05 2017