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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
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
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
Sequence in context: A233058 A301721 A176449 * A327628 A169877 A286710
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Dec 02 2016
STATUS
approved

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Last modified July 23 21:46 EDT 2024. Contains 374575 sequences. (Running on oeis4.)