A278947 - OEIS

%I #30 Dec 30 2024 02:16:06

%S 0,1,8,19,42,56,107,113,190,208,298,287,483,404,589,614,806,698,1079,

%T 875,1302,1202,1471,1289,2035,1581,2062,1990,2541,2060,3142,2357,3318,

%U 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

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

%C Inverse Moebius transform of heptagonal numbers (A000566).

%H Seiichi Manyama, <a href="/A278947/b278947.txt">Table of n, a(n) for n = 0..10000</a>

%H Mira Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H Mira Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>.

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.

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

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

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

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

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

%t nmax=58; CoefficientList[Series[Sum[(k (5 k - 3)/2) x^k/(1 - x^k),  {k, 1, nmax}], {x, 0, nmax}], x]

%t Flatten[{0, Table[(5*DivisorSigma[2, n] - 3*DivisorSigma[1, n])/2, {n, 1, 100}]}] (* _Vaclav Kotesovec_, Dec 05 2016 *)

%o (PARI) a(n) = if(n == 0, 0, my(f = factor(n)); (5 * sigma(f, 2) - 3 * sigma(f)) / 2); \\ _Amiram Eldar_, Dec 29 2024

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

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

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Dec 02 2016