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%I #37 Oct 31 2023 04:44:47
%S 0,1,144,2268,18688,78750,326592,825944,2396160,4966677,11340000,
%T 19501812,42384384,62777078,118935936,178605000,306774016,410422194,
%U 715201488,894002060,1471680000,1873240992,2808260928,3405105288,5434490880,6152734375,9039899232
%N Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C Multiplicative because A001158 is. - _Andrew Howroyd_, Jul 23 2018
%H Seiichi Manyama, <a href="/A280025/b280025.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = n^4*A001158(n) for n > 0.
%F a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
%F Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - _Amiram Eldar_, Dec 08 2022
%F From _Amiram Eldar_, Oct 31 2023: (Start)
%F Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
%F Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
%t Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* _Amiram Eldar_, Oct 31 2023 *)
%o (PARI) a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ _Andrew Howroyd_, Jul 23 2018
%Y Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
%Y 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 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 Cf. A152649.
%K nonn,easy,mult
%O 0,3
%A _Seiichi Manyama_, Feb 22 2017