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The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.
2

%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