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%I #33 Oct 30 2023 14:37:20
%S 0,1,3,7,10,19,21,35,39,56,55,91,78,113,118,155,136,208,171,252,234,
%T 287,253,395,310,404,390,497,406,614,465,651,586,698,626,910,666,875,
%U 822,1060,820,1202,903,1239,1144,1289,1081,1643,1197,1581,1414,1736
%N a(n) = Sum_{d|n} d*(d-1)/2.
%C Inverse Mobius transform of A000217. - _R. J. Mathar_, Jan 19 2009
%H Seiichi Manyama, <a href="/A069153/b069153.txt">Table of n, a(n) for n = 1..10000</a>
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%F G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - _Vladeta Jovovic_, Dec 17 2002
%F Row sums of triangle A134840. - _Gary W. Adamson_, Nov 12 2007
%F G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - _Michael Somos_, Feb 12 2008
%F G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - _Michael Somos_, Feb 12 2008
%F From _Peter Bala_, Jan 21 2021: (Start)
%F a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n) A000203(n)) = (1/2)*A086666.
%F G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
%e x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...
%p with(numtheory):
%p seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # _Peter Bala_, Jan 21 2021
%t A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* _Enrique Pérez Herrero_, Feb 21 2012 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}
%Y Cf. A000203, A001157, A007437, A086666, A134840.
%Y Cf. A032741, A065608, A363604, A363605, A363606.
%K easy,nonn
%O 1,3
%A _Benoit Cloitre_, Apr 08 2002
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