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The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.
2

%I #11 May 02 2018 06:12:16

%S 2,4,10,17,30,44,67,91,126,163,213,265,333,403,491,582,693,807,944,

%T 1084,1249,1418,1614,1814,2044,2278,2544,2815,3120,3430,3777,4129,

%U 4520,4917,5355,5799,6287,6781,7321,7868,8463,9065,9718,10378,11091,11812,12588,13372,14214,15064

%N The number of trees with 4 nodes labeled by positive integers, where each tree's label sum is n.

%C Computed by the sum over the A000055(4)=2 shapes of the trees: the linear graph of the n-Butane, and the star graph of (1)-Methyl-Propane.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,-1,2,1,-1).

%F a(n) = A005993(n-4)+A000601(n-4).

%F G.f.: x^4*(2+2*x+2*x^2+x^3+x^4)/((1+x)^2*(x-1)^4*(1+x+x^2) ).

%e a(4)=2 because there is a linear tree with all labels equal 1 and the star tree with all labels equal to 1.

%p x^4*(2+2*x+2*x^2+x^3+x^4)/(1+x)^2/(x-1)^4/(1+x+x^2) ;

%p taylor(%,x=0,80) ;

%p gfun[seriestolist](%) ;

%Y 4th column of A303841.

%K nonn,easy

%O 4,1

%A _R. J. Mathar_, Mar 26 2018