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A282213 Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

%I A282213 #41 Aug 02 2025 07:30:40
%S A282213 0,1,72,756,4672,15750,54432,117992,299520,551853,1134000,1772892,
%T A282213 3532032,4829006,8495424,11907000,19173376,24142482,39733416,47052740,
%U A282213 73584000,89201952,127648224,148048056,226437120,246109375,347688432,402320520,551258624,594847710
%N A282213 Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C A282213 Multiplicative because A001158 is. - _Andrew Howroyd_, Jul 25 2018
%H A282213 Seiichi Manyama, <a href="/A282213/b282213.txt">Table of n, a(n) for n = 0..1000</a>
%F A282213 G.f.: phi_{6, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F A282213 a(n) = (A282586(n) - 3*A282595(n) + 3*A282101(n) - A013974(n))/3456. - _Seiichi Manyama_, Feb 19 2017
%F A282213 a(n) = n^3*A001158(n) for n > 0. - _Seiichi Manyama_, Feb 19 2017
%F A282213 Sum_{k=1..n} a(k) ~ zeta(4) * n^7 / 7. - _Amiram Eldar_, Sep 06 2023
%F A282213 From _Amiram Eldar_, Oct 31 2023: (Start)
%F A282213 Multiplicative with a(p^e) = p^(3*e) * (p^(3*e+3)-1)/(p^3-1).
%F A282213 Dirichlet g.f.: zeta(s-3)*zeta(s-6). (End)
%F A282213 G.f.: Sum_{k>=1} k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - _Vaclav Kotesovec_, Aug 02 2025
%e A282213 a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.
%t A282213 terms = 30;
%t A282213 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A282213 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A282213 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A282213 (E2[x]^3*E4[x] - 3 E2[x]^2*E6[x] + 3 E2[x] E4[x]^2 - E4[x] E6[x])/3456 + O[x]^terms // CoefficientList[#, x]&
%t A282213 (* or: *)
%t A282213 Table[n^3*DivisorSigma[3, n], {n, 0, terms-1}] (* _Jean-François Alcover_, Feb 27 2018 *)
%t A282213 nmax = 30; CoefficientList[Series[Sum[k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A282213 (PARI) a(n) = if (n, n^3*sigma(n, 3), 0); \\ _Michel Marcus_, Feb 27 2018
%Y A282213 Cf. A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).
%Y A282213 Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282586 (E_2^3*E_4), A282595 (E_2^2*E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
%Y A282213 Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n))
%Y A282213 Cf. A013662.
%K A282213 nonn,easy,mult
%O A282213 0,3
%A A282213 _Seiichi Manyama_, Feb 09 2017