%I #11 Apr 16 2018 05:00:24
%S 3,9,24,50,96,164,267,408,603,856,1186,1598,2115,2742,3505,4411,5489,
%T 6746,8215,9904,11849,14059,16573,19401,22586,26138,30103,34493,39357,
%U 44707,50596,57037,64086,71757,80109,89157,98964,109545,120966,133244,146448,160595,175758,191955
%N The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.
%C Computed by the sum over the A000055(5)=3 shapes of the trees: the linear graph of the n-Pentane, the branched 2-Methyl-Butane, and the star graph of (1,1)-Bimethyl-Propane.
%H R. J. Mathar, <a href="/A301740/a301740_1.pdf">Labeled Trees with Fixed Node Label sum</a>, sequence v_5.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,-2,2,0,1,0,-2,1).
%F a(n) = A005994(n-5)+A001752(n-5)+A002621(n-5).
%e a(5)=3 because there is a linear tree with all labels equal 1, the branched tree with all labels equal to 1, and the star tree with all labels equal 1.
%p -x^5*(3+3*x+6*x^2+5*x^3+5*x^4+2*x^5+x^6)/(1+x^2)/(1+x+x^2)/(1+x)^2/(x-1)^5 ;
%p taylor(%,x=0,80) ;
%p gfun[seriestolist](%) ;
%Y Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).
%K nonn,easy
%O 5,1
%A _R. J. Mathar_, Mar 26 2018