|
%I #37 Oct 30 2023 01:52:32
%S 0,1,36,252,1168,3150,9072,16856,37440,61317,113400,161172,294336,
%T 371462,606816,793800,1198336,1420146,2207412,2476460,3679200,4247712,
%U 5802192,6436872,9434880,9844375,13372632,14900760,19687808,20511990,28576800,28630112,38347776
%N Coefficients in q-expansion of (E_2^2*E_4 - 2*E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C Multiplicative because A001158 is. - _Andrew Howroyd_, Jul 25 2018
%H Seiichi Manyama, <a href="/A282099/b282099.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: phi_{5, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F a(n) = (A282208(n) - 2*A282096(n) + A008410(n))/1728. - _Seiichi Manyama_, Feb 19 2017
%F a(n) = n^2*A001158(n) for n > 0. - _Seiichi Manyama_, Feb 19 2017
%F Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 540. - _Vaclav Kotesovec_, May 09 2022
%F From _Amiram Eldar_, Oct 30 2023: (Start)
%F Multiplicative with a(p^e) = p^(2*e) * (p^(3*e+3)-1)/(p^3-1).
%F Dirichlet g.f.: zeta(s-2)*zeta(s-5). (End)
%e a(6) = 1^5*6^2 + 2^5*3^2 + 3^5*2^2 + 6^5*1^2 = 9072.
%t a[0]=0;a[n_]:=(n^2)*DivisorSigma[3,n];Table[a[n],{n,0,32}] (* _Indranil Ghosh_, Feb 21 2017 *)
%o (PARI) a(n) = if (n==0, 0, n^2*sigma(n, 3)); \\ _Michel Marcus_, Feb 21 2017
%Y Cf. A282097 (phi_{3, 2}), this sequence (phi_{5, 2}).
%Y Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_8 = E_4^2).
%Y Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), this sequence (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)).
%K nonn,easy,mult
%O 0,3
%A _Seiichi Manyama_, Feb 06 2017
|
