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A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

%I A328490 #20 Nov 02 2024 11:48:54
%S A328490 1,10,20,67,52,200,100,380,282,520,244,1340,340,1000,1040,1973,580,
%T A328490 2820,724,3484,2000,2440,1060,7600,1978,3400,3460,6700,1684,10400,
%U A328490 1924,9710,4880,5800,5200,18894,2740,7240,6800,19760,3364,20000,3700,16348,14664
%N A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.
%C A328490 Dirichlet convolution of A001157 with itself.
%C A328490 Dirichlet convolution of A000005 with A034714.
%C A328490 Dirichlet convolution of A000290 with A007433.
%F A328490 a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.
%F A328490 a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.
%F A328490 Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 17 2019
%F A328490 Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - _Amiram Eldar_, Sep 15 2023
%t A328490 Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
%t A328490 f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2023 *)
%o A328490 (Magma) [&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // _Marius A. Burtea_, Oct 16 2019
%o A328490 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Sep 26 2020
%Y A328490 Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.
%Y A328490 Cf. A175705.
%K A328490 nonn,easy,mult
%O A328490 1,2
%A A328490 _Ilya Gutkovskiy_, Oct 16 2019