This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A301740 #13 Jun 16 2025 12:54:37
%S A301740 3,9,24,50,96,164,267,408,603,856,1186,1598,2115,2742,3505,4411,5489,
%T A301740 6746,8215,9904,11849,14059,16573,19401,22586,26138,30103,34493,39357,
%U A301740 44707,50596,57037,64086,71757,80109,89157,98964,109545,120966,133244,146448,160595,175758,191955
%N A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.
%C A301740 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 A301740 R. J. Mathar, <a href="/A301740/a301740_1.pdf">Labeled Trees with Fixed Node Label sum</a>, sequence v_5.
%H A301740 <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 A301740 a(n) = A005994(n-5)+A001752(n-5)+A002621(n-5).
%e A301740 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 A301740 -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 A301740 taylor(%,x=0,80) ;
%p A301740 gfun[seriestolist](%) ;
%t A301740 LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{3,9,24,50,96,164,267,408,603,856,1186},50] (* _Harvey P. Dale_, Jun 16 2025 *)
%Y A301740 Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).
%K A301740 nonn,easy
%O A301740 5,1
%A A301740 _R. J. Mathar_, Mar 26 2018