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A281372 Coefficients in q-expansion of (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 A281372 #58 Jan 22 2024 10:12:41
%S A281372 0,1,18,84,292,630,1512,2408,4680,6813,11340,14652,24528,28574,43344,
%T A281372 52920,74896,83538,122634,130340,183960,202272,263736,279864,393120,
%U A281372 393775,514332,551880,703136,707310,952560,923552,1198368,1230768,1503684,1517040,1989396,1874198,2346120,2400216,2948400
%N A281372 Coefficients in q-expansion of (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 A281372 The q-expansion of the square of this expression is given in A281371.
%C A281372 Multiplicative because A001158 is. - _Andrew Howroyd_, Jul 23 2018
%H A281372 Seiichi Manyama, <a href="/A281372/b281372.txt">Table of n, a(n) for n = 0..1000</a>
%F A281372 a(n) = A145094(n)/240 for n > 0. - _Seiichi Manyama_, Feb 04 2017
%F A281372 G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - _Seiichi Manyama_, Feb 04 2017
%F A281372 a(n) = n*A001158(n) for n > 0. - _Seiichi Manyama_, Feb 18 2017
%F A281372 G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Aug 31 2017
%F A281372 Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 450. - _Vaclav Kotesovec_, May 09 2022
%F A281372 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A281372 Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
%F A281372 Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
%F A281372 a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - _Peter Bala_, Jan 19 2024
%F A281372 a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - _Peter Bala_, Jan 22 2024
%p A281372 with(gfun):
%p A281372 with(numtheory); M:=100;
%p A281372 E := proc(k) local n, t1; global M;
%p A281372 t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
%p A281372 series(t1, q, M+1); end;
%p A281372 e2:=E(2); e4:=E(4); e6:=E(6);
%p A281372 t1:=series((e2*e4-e6)/720,q,M+1);
%p A281372 seriestolist(t1);
%p A281372 # alternative program
%p A281372 seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # _Peter Bala_, Jan 20 2024
%t A281372 Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* _Indranil Ghosh_, Mar 11 2017 *)
%t A281372 terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* _Jean-François Alcover_, Mar 01 2018 *)
%o A281372 (PARI) for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ _Indranil Ghosh_, Mar 11 2017
%o A281372 (Magma) [0] cat [n*DivisorSigma(3, n): n in [1..50]]; // _Vincenzo Librandi_, Mar 01 2018
%Y A281372 Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.
%K A281372 nonn,easy,mult
%O A281372 0,3
%A A281372 _N. J. A. Sloane_, Feb 04 2017