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A083413 a(n) = Sum_{d|n} d*2^(d-1) for n > 0.

%I A083413 #16 May 21 2018 08:48:35
%S A083413 0,1,5,13,37,81,209,449,1061,2317,5205,11265,24817,53249,115141,
%T A083413 245853,525349,1114113,2361809,4980737,10490997,22020557,46148613,
%U A083413 96468993,201352433,419430481,872468485,1811941645,3758211557,7784628225,16106378529,33285996545
%N A083413 a(n) = Sum_{d|n} d*2^(d-1) for n > 0.
%H A083413 N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003.
%H A083413 N. J. A. Sloane and Thomas Wieder, <a href="https://doi.org/10.1007/s11083-004-9460-9">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.
%F A083413 Sum_{n > 0} a(n)*x^n/n = Sum_{m > 0} x^m/(m*(1-2*x^m)).
%F A083413 G.f.: Sum_{m > 0} x^m/(1-2*x^m)^2.
%F A083413 a(n) ~ n*2^(n-2). - _Vaclav Kotesovec_, Sep 09 2014
%F A083413 L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2^(k-1))) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 20 2018
%p A083413 oo := 101: t1 := add(x^m/(m*(1-2*x^m)),m=1..oo): series(%,x,oo): t2 := seriestolist(%): A083413 := n -> t2[n+1]*n;
%t A083413 CoefficientList[Series[Sum[x^k/(1-2*x^k)^2,{k,1,30}],{x,0,30}],x] (* _Vaclav Kotesovec_, Sep 09 2014 *)
%o A083413 (PARI) a(n)=if(n<1,0,sumdiv(n,d,d*2^(d-1)))
%Y A083413 Cf. A077272.
%Y A083413 Cf. A054599.
%K A083413 nonn
%O A083413 0,3
%A A083413 _N. J. A. Sloane_, Jun 09 2003