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A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A280022 #39 Aug 02 2025 10:46:26
%S A280022 0,1,48,324,1792,3750,15552,19208,61440,85293,180000,175692,580608,
%T A280022 399854,921984,1215000,2031616,1503378,4094064,2606420,6720000,
%U A280022 6223392,8433216,6716184,19906560,12109375,19192992,21257640,34420736,21218430,58320000,29552672
%N A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A280022 Multiplicative because A000203 is. - _Andrew Howroyd_, Jul 23 2018
%H A280022 Seiichi Manyama, <a href="/A280022/b280022.txt">Table of n, a(n) for n = 0..1000</a>
%F A280022 a(n) = n^4*A000203(n) for n > 0.
%F A280022 a(n) = (15*A282101(n) - 20*A282595(n) + 10*A282586(n) - 4*A013974(n) - A282431(n))/20736.
%F A280022 Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^2/36 = 0.274155... (A353908). - _Amiram Eldar_, Dec 08 2022
%F A280022 From _Amiram Eldar_, Oct 31 2023: (Start)
%F A280022 Multiplicative with a(p^e) = p^(4*e) * (p^(e+1)-1)/(p-1).
%F A280022 Dirichlet g.f.: zeta(s-4)*zeta(s-5). (End)
%F A280022 G.f.: Sum_{k>=1} k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6. - _Vaclav Kotesovec_, Aug 02 2025
%t A280022 Table[n^4 * DivisorSigma[1, n], {n, 0, 32}] (* _Amiram Eldar_, Oct 31 2023 *)
%t A280022 nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A280022 (PARI) a(n) = if(n < 1, 0, n^4 * sigma(n)); \\ _Andrew Howroyd_, Jul 23 2018
%Y A280022 Cf. this sequence (phi_{5, 4}), A280025 (phi_{7, 4}).
%Y A280022 Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A282586 (E_2^3*E_4), A013974 (E_4*E_6 = E_10), A282431 (E_2^5).
%Y A280022 Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), A282211 (n^3*sigma(n)), this sequence (n^4*sigma(n)).
%Y A280022 Cf. A353908.
%K A280022 nonn,easy,mult
%O A280022 0,3
%A A280022 _Seiichi Manyama_, Feb 22 2017