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A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A280025 #40 Aug 02 2025 10:39:48
%S A280025 0,1,144,2268,18688,78750,326592,825944,2396160,4966677,11340000,
%T A280025 19501812,42384384,62777078,118935936,178605000,306774016,410422194,
%U A280025 715201488,894002060,1471680000,1873240992,2808260928,3405105288,5434490880,6152734375,9039899232
%N A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A280025 Multiplicative because A001158 is. - _Andrew Howroyd_, Jul 23 2018
%H A280025 Seiichi Manyama, <a href="/A280025/b280025.txt">Table of n, a(n) for n = 0..1000</a>
%F A280025 a(n) = n^4*A001158(n) for n > 0.
%F A280025 a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
%F A280025 Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - _Amiram Eldar_, Dec 08 2022
%F A280025 From _Amiram Eldar_, Oct 31 2023: (Start)
%F A280025 Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
%F A280025 Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
%F A280025 G.f.: Sum_{k>=1} k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8. - _Vaclav Kotesovec_, Aug 02 2025
%t A280025 Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* _Amiram Eldar_, Oct 31 2023 *)
%t A280025 nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A280025 (PARI) a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ _Andrew Howroyd_, Jul 23 2018
%Y A280025 Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
%Y A280025 Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
%Y A280025 Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).
%Y A280025 Cf. A152649.
%K A280025 nonn,easy,mult
%O A280025 0,3
%A A280025 _Seiichi Manyama_, Feb 22 2017