A278945 Expansion of Sum_{k>=1} k*(2*k - 1)*x^k/(1 - x^k).

0, 1, 7, 16, 35, 46, 88, 92, 155, 169, 242, 232, 392, 326, 476, 496, 651, 562, 871, 704, 1050, 968, 1184, 1036, 1640, 1271, 1658, 1600, 2044, 1654, 2528, 1892, 2667, 2392, 2846, 2552, 3731, 2702, 3560, 3344, 4330, 3322, 4904, 3656, 5040, 4654, 5228, 4372, 6696, 4845, 6417, 5728, 7042, 5566, 8080, 6272, 8380, 7160, 8330, 6904, 10752

Offset: 0

Author

Ilya Gutkovskiy, Dec 02 2016

Keywords

• nonn
• easy

Comments

Inverse Moebius transform of hexagonal numbers (A000384).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms.
• Eric Weisstein's World of Mathematics, Hexagonal Number.
• Index to sequences related to polygonal numbers.

Crossrefs

Cf. A000203, A000384, A001157, A002117, A007437, A059358, A116913.

Programs

Magma

[0] cat [2*DivisorSigma(2, n) - DivisorSigma(1, n): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016

Mathematica

nmax=60; CoefficientList[Series[Sum[k (2 k - 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Flatten[{0, Table[2*DivisorSigma[2, n] - DivisorSigma[1, n], {n, 1, 100}]}] (* Vaclav Kotesovec, Dec 05 2016 *)

PARI

a(n) = if(n == 0, 0, my(f = factor(n)); 2 * sigma(f, 2) - sigma(f)); \\ Amiram Eldar, Dec 29 2024

Formula

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

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

a(n) = Sum_{d|n} d*(2*d - 1).

a(n) = 2*A001157(n) - A000203(n).

Sum_{k=1..n} a(k) ~ (2*zeta(3)/3) * n^3. - Amiram Eldar, Dec 29 2024