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A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

%I A282050 #45 Aug 01 2025 09:33:12
%S A282050 0,1,66,732,4228,15630,48312,117656,270600,533637,1031580,1771572,
%T A282050 3094896,4826822,7765296,11441160,17318416,24137586,35220042,47045900,
%U A282050 66083640,86124192,116923752,148035912,198079200,244218775,318570252,389021400,497449568,594823350
%N A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C A282050 Multiplicative because A001160 is. - _Andrew Howroyd_, Jul 23 2018
%H A282050 Seiichi Manyama, <a href="/A282050/b282050.txt">Table of n, a(n) for n = 0..1000</a>
%F A282050 a(n) = A145095(n)/504 for n > 0.
%F A282050 G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F A282050 a(n) = (A008410(n) - A282096(n))/1008. - _Seiichi Manyama_, Feb 10 2017
%F A282050 If p is a prime, a(p) = p^6 + p = A131472(p). - _Seiichi Manyama_, Feb 10 2017
%F A282050 a(n) = n*A001160(n) for n > 0. - _Seiichi Manyama_, Feb 18 2017
%F A282050 Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - _Amiram Eldar_, Sep 06 2023
%F A282050 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282050 Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
%F A282050 Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
%F A282050 G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - _Vaclav Kotesovec_, Aug 01 2025
%e A282050 a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.
%t A282050 terms = 30;
%t A282050 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A282050 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A282050 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A282050 (E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%t A282050 Table[n * DivisorSigma[5, n], {n, 0, 30}] (* _Amiram Eldar_, Sep 06 2023 *)
%t A282050 nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 01 2025 *)
%o A282050 (PARI) a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ _Andrew Howroyd_, Jul 23 2018
%Y A282050 Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
%Y A282050 Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
%Y A282050 Cf. A001160, A013664, A131472.
%K A282050 nonn,easy,mult
%O A282050 0,3
%A A282050 _Seiichi Manyama_, Feb 05 2017