A278947 Expansion of Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).

0, 1, 8, 19, 42, 56, 107, 113, 190, 208, 298, 287, 483, 404, 589, 614, 806, 698, 1079, 875, 1302, 1202, 1471, 1289, 2035, 1581, 2062, 1990, 2541, 2060, 3142, 2357, 3318, 2978, 3544, 3178, 4641, 3368, 4435, 4166, 5390, 4142, 6106, 4559, 6279, 5798, 6517, 5453, 8339, 6042, 7998, 7142, 8778, 6944, 10070, 7822, 10445, 8930, 10390

Offset: 0

Author

Ilya Gutkovskiy, Dec 02 2016

Keywords

• nonn
• easy

Comments

Inverse Moebius transform of heptagonal numbers (A000566).

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, Heptagonal Number.
• Index to sequences related to polygonal numbers.

Crossrefs

Cf. A000203, A000566 (heptagonal numbers), A002117, A059358.

Inverse Moebius transforms of polygonal numbers: A007437 (k=3), A001157 (k=4), A116913 (k=5), A278945 (k=6), this sequence (k=7).

Programs

Mathematica

nmax=58; CoefficientList[Series[Sum[(k (5 k - 3)/2) x^k/(1 - x^k),  {k, 1, nmax}], {x, 0, nmax}], x]
Flatten[{0, Table[(5*DivisorSigma[2, n] - 3*DivisorSigma[1, n])/2, {n, 1, 100}]}] (* Vaclav Kotesovec, Dec 05 2016 *)

PARI

a(n) = if(n == 0, 0, my(f = factor(n)); (5 * sigma(f, 2) - 3 * sigma(f)) / 2); \\ Amiram Eldar, Dec 29 2024

Formula

G.f.: Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).

Dirichlet g.f.: (5*zeta(s-2) - 3*zeta(s-1))*zeta(s)/2.

a(n) = Sum_{d|n} d*(5*d - 3)/2.

a(n) = (5*A001157(n) - 3*A000203(n))/2.

Sum_{k=1..n} a(k) ~ (5*zeta(3)/6) * n^3. - Amiram Eldar, Dec 29 2024