A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

1, 5, 11, 24, 32, 61, 65, 109, 120, 172, 167, 279, 236, 343, 358, 470, 410, 630, 515, 762, 706, 865, 761, 1193, 933, 1216, 1174, 1497, 1220, 1850, 1397, 1959, 1762, 2098, 1882, 2739, 2000, 2629, 2470, 3188, 2462, 3614, 2711, 3723, 3438, 3871, 3245, 4939, 3594, 4749, 4246, 5214

Offset: 1

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

• nonn
• easy

Comments

Inverse Moebius transform of centered triangular numbers (A005448).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A001157, A002117, A005448, A007437, A059358.

Programs

Mathematica

nmax = 52; CoefficientList[Series[Sum[x^k (1 - x^(3 k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[3 (DivisorSigma[2, n] - DivisorSigma[1, n])/2 + DivisorSigma[0, n], {n, 1, 52}]

PARI

a(n)={sumdiv(n, d, 3*d*(d-1)/2 + 1)} \\ Andrew Howroyd, Aug 14 2019

PARI

a(n)={3*(sigma(n,2) - sigma(n))/2 + numdiv(n)} \\ Andrew Howroyd, Aug 14 2019

Formula

G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^k/(1 - x^k).

a(n) = 3 * (sigma_2(n) - sigma_1(n))/2 + d(n).

From Amiram Eldar, Jan 02 2025: (Start)

Dirichlet g.f.: zeta(s) * (3 * zeta(s-2) - 3 * zeta(s-1) + 2 * zeta(s)) / 2.

Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)