A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 516, 19692, 264208, 1953150, 10161072, 40353656, 135274560, 387597717, 1007825400, 2357947812, 5202783936, 10604499542, 20822486496, 38461429800, 69260574976, 118587876786, 200000421972, 322687698140, 516037855200, 794644193952, 1216701070992

Offset: 0

Author

Seiichi Manyama, Feb 21 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).

Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).

Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).

Cf. A013666.

Programs

Mathematica

Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* Indranil Ghosh, Mar 12 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

PARI

for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ Indranil Ghosh, Mar 12 2017

Formula

a(n) = n^2*A013955(n) for n > 0.

a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.

Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)

G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025