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A278945 Expansion of Sum_{k>=1} k*(2*k - 1)*x^k/(1 - x^k). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Inverse Moebius transform of hexagonal numbers (A000384).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. 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]

M. 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

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).

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A233058 A301721 A176449 * A327628 A169877 A286710

Adjacent sequences:  A278942 A278943 A278944 * A278946 A278947 A278948

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Dec 02 2016

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)