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%I #51 Jan 27 2024 05:40:01
%S 0,1,12,36,112,150,432,392,960,1053,1800,1452,4032,2366,4704,5400,
%T 7936,5202,12636,7220,16800,14112,17424,12696,34560,19375,28392,29160,
%U 43904,25230,64800,30752,64512,52272,62424,58800,117936,52022,86640,85176,144000,70602
%N Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
%C Multiplicative because A000203 is. - _Andrew Howroyd_, Jul 25 2018
%H Seiichi Manyama, <a href="/A282097/b282097.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
%F G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%F a(n) = n^2*A000203(n) for n > 0. - _Seiichi Manyama_, Feb 19 2017
%F G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - _Ilya Gutkovskiy_, May 02 2018
%F From _Amiram Eldar_, Oct 30 2023: (Start)
%F Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
%F Dirichlet g.f.: zeta(s-2)*zeta(s-3).
%F Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
%F From _Peter Bala_, Jan 22 2024:
%F a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
%F a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
%F a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)
%e a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.
%t a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* _Indranil Ghosh_, Feb 21 2017 *)
%t terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* _Jean-François Alcover_, Mar 01 2018 *)
%o (PARI) a(n) = if (n==0, 0, n^2*sigma(n)); \\ _Michel Marcus_, Feb 21 2017
%o (Magma) [0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // _Vincenzo Librandi_, Mar 01 2018
%Y Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
%Y Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
%Y Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
%Y Cf. A222171.
%K nonn,easy,mult
%O 0,3
%A _Seiichi Manyama_, Feb 06 2017
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