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

0, 1, 132, 2196, 16912, 78150, 289872, 823592, 2164800, 4802733, 10315800, 19487292, 37138752, 62748686, 108714144, 171617400, 277094656, 410338962, 633960756, 893872100, 1321672800, 1808608032, 2572322544, 3404825976, 4753900800, 6105469375, 8282826552

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. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), this sequence (phi_{7, 2}), A282753 (phi_{9, 2}).

Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A013974 (E_4*E_6 = E_10).

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

Cf. A013664.

Programs

Mathematica

Table[n^2 * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
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 *)

PARI

a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n^2*A001160(n) for n > 0.

a(n) = (2*A282101(n) - A282595(n) - A013974(n))/1728.

Sum_{k=1..n} a(k) ~ zeta(6) * n^8 / 8. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(5*e+5)-1)/(p^5-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-7). (End)

G.f.: Sum_{k>=1} k^7*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025