A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.

G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).

From Amiram Eldar, Dec 30 2024: (Start)

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

Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

Mathematica

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

Prog

(PARI) my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))
(PARI) a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

Crossrefs

Cf. A000292, A032741, A065608, A069153, A363605, A363606.

Cf. A059358, A363607, A363611.

Cf. A000203, A001158, A013662, A092348.

Sequence in context: A301075 A038413 A008174 * A008262 A075099 A240784

Adjacent sequences: A363601 A363602 A363603 * A363605 A363606 A363607

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 11 2023

Status

approved