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%I #9 Feb 02 2019 06:49:13
%S 1,9,31,89,176,375,589,1049,1516,2384,3147,4823,5916,8437,10406,14105,
%T 16474,22380,25271,33264,37810,47683,52901,68183,73301,91100,100174,
%U 122197,130356,161750,169137,205593,219162,259242,272714,330524,338144,400719,421686,493424
%N a(n) = Sum_{d|n} d*binomial(d+2,3).
%C Inverse Möbius transform of A002417.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Sum_{k>=1} x^k*(1 + 3*x^k)/(1 - x^k)^5.
%F G.f.: Sum_{k>=1} k*A000292(k)*x^k/(1 - x^k).
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^A000292(k)) = Sum_{n>=1} a(n)*x^n/n.
%F Dirichlet g.f.: (zeta(s-4) + 3*zeta(s-3) + 2*zeta(s-2))*zeta(s)/6.
%F a(n) = (2*sigma_2(n) + 3*sigma_3(n) + sigma_4(n))/6.
%F a(n) = Sum_{d|n} A002417(d).
%F Sum_{k=1..n} a(k) ~ Zeta(5) * n^5 / 30. - _Vaclav Kotesovec_, Feb 02 2019
%t Table[Sum[d Binomial[d + 2, 3], {d, Divisors[n]}], {n, 40}]
%t nmax = 40; Rest[CoefficientList[Series[Sum[x^k (1 + 3 x^k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[(2 DivisorSigma[2, n] + 3 DivisorSigma[3, n] + DivisorSigma[4, n])/6, {n, 40}]
%Y Cf. A000292, A000335, A001157, A001158, A001159, A002417, A059358, A278403.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Nov 14 2018
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