A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

1, 6, 15, 36, 56, 111, 141, 240, 300, 446, 507, 791, 820, 1161, 1310, 1736, 1786, 2505, 2471, 3346, 3466, 4307, 4325, 5895, 5581, 7026, 7230, 8905, 8556, 11246, 10417, 13176, 13050, 15476, 15106, 19391, 17576, 21495, 21374, 25690, 23822, 30162, 27435, 33707, 32990, 37841, 35721

Offset: 1

Author

Ilya Gutkovskiy, Oct 24 2018

Keywords

• nonn
• easy

Comments

Inverse Möbius transform of square pyramidal numbers (A000330).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000203, A000330, A001157, A001158, A007437, A059358.

Cf. A135539.

Programs

Maple

a:=series(add(x^k*(1+x^k)/(1-x^k)^4,k=1..100),x=0,48): seq(coeff(a,x,n),n=1..47); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]
Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]

PARI

a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 2) + sigma(f, 1)) / 6; \\ Amiram Eldar, Jan 03 2025

Formula

G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.

a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.

a(n) = Sum_{i=1..n} i^2*A135539(n,i). - Ridouane Oudra, Jul 22 2022

From Amiram Eldar, Jan 03 2025: (Start)

Dirichlet g.f.: zeta(s) * (2*zeta(s-3) + 3*zeta(s-2) + zeta(s-1)) / 6.

Sum_{k=1..n} a(k) ~ (zeta(4)/12) * n^4. (End)