A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 264, 6588, 67648, 390750, 1739232, 5765144, 17318400, 43224597, 103158000, 214360212, 445665024, 815732918, 1521998016, 2574261000, 4433514496, 6975762354, 11411293608, 16983569900, 26433456000, 37980768672, 56591095968, 78310997448

Offset: 0

Author

Seiichi Manyama, Feb 21 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A001160 is. - Andrew Howroyd, Jul 25 2018

Links

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

Crossrefs

Cf. A282211 (phi_{4, 3}), A282213 (phi_{6, 3}), this sequence (phi_{8, 3}).

Cf. A282752 (E_2^2*E_4^2), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).

Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), A282751 (n^2*sigma_5(n)), this sequence (n^3*sigma_5(n)).

Cf. A013664.

Programs

Mathematica

a[0]=0;a[n_]:=(n^3)*DivisorSigma[5,n];Table[a[n],{n,0,23}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 30; CoefficientList[Series[Sum[k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

PARI

a(n) = if (n==0, 0, n^3*sigma(n, 5)); \\ Michel Marcus, Feb 21 2017

Formula

a(n) = n^3*A001160(n) for n > 0.

a(n) = (6*A282752(n) - 2*A282780(n) - 6*A282102(n) + A008411(n) + A280869(n))/5184.

Sum_{k=1..n} a(k) ~ zeta(6) * n^9 / 9. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 31 2023: (Start)

Multiplicative with a(p^e) = p^(3*e) * (p^(5*e+5)-1)/(p^5-1).

Dirichlet g.f.: zeta(s-3)*zeta(s-8). (End)

G.f.: Sum_{k>=1} k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Vaclav Kotesovec, Aug 02 2025