cp's OEIS Frontend

A282751 Expansion of phi_{7, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A282751 #25 Aug 02 2025 03:18:09
%S A282751 0,1,132,2196,16912,78150,289872,823592,2164800,4802733,10315800,
%T A282751 19487292,37138752,62748686,108714144,171617400,277094656,410338962,
%U A282751 633960756,893872100,1321672800,1808608032,2572322544,3404825976,4753900800,6105469375,8282826552
%N A282751 Expansion of phi_{7, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A282751 Multiplicative because A001160 is. - _Andrew Howroyd_, Jul 25 2018
%H A282751 Seiichi Manyama, <a href="/A282751/b282751.txt">Table of n, a(n) for n = 0..1000</a>
%F A282751 a(n) = n^2*A001160(n) for n > 0.
%F A282751 a(n) = (2*A282101(n) - A282595(n) - A013974(n))/1728.
%F A282751 Sum_{k=1..n} a(k) ~ zeta(6) * n^8 / 8. - _Amiram Eldar_, Sep 06 2023
%F A282751 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282751 Multiplicative with a(p^e) = p^(2*e) * (p^(5*e+5)-1)/(p^5-1).
%F A282751 Dirichlet g.f.: zeta(s-2)*zeta(s-7). (End)
%F A282751 G.f.: Sum_{k>=1} k^7*x^k*(1 + x^k)/(1 - x^k)^3. - _Vaclav Kotesovec_, Aug 02 2025
%t A282751 Table[n^2 * DivisorSigma[5, n], {n, 0, 30}] (* _Amiram Eldar_, Sep 06 2023 *)
%t A282751 nmax = 40; CoefficientList[Series[Sum[k^7*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%o A282751 (PARI) a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ _Andrew Howroyd_, Jul 25 2018
%Y A282751 Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), this sequence (phi_{7, 2}), A282753 (phi_{9, 2}).
%Y A282751 Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A013974 (E_4*E_6 = E_10).
%Y A282751 Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), this sequence (n^2*sigma_5(n)).
%Y A282751 Cf. A013664.
%K A282751 nonn,easy,mult
%O A282751 0,3
%A A282751 _Seiichi Manyama_, Feb 21 2017