%I #26 Nov 03 2023 15:49:35
%S 1,2,15,144,1765,26400,466459,9508352,219651849,5671088640,
%T 161833149511,5058050224128,171837337744813,6304955850432512,
%U 248477268083174355,10467916801317273600,469451601966727952401,22329535184262444220416,1122809130124800181976575
%N Total number of even outdegree nodes among all labeled rooted trees on n nodes.
%H Alois P. Heinz, <a href="/A253571/b253571.txt">Table of n, a(n) for n = 1..380</a> (first 100 terms from Marko Riedel)
%H Marko Riedel, <a href="http://math.stackexchange.com/questions/551427/">Even outdegree nodes among all labeled trees on n nodes</a>
%F E.g.f.: (T^2+z^2)/(2*T*(1-T)) where T is the labeled tree function defined by T = z exp T.
%e When n=3 there are two types of trees: rooted paths on three nodes which have one even degree node (the bottom one with zero children), giving 6*1, and trees consisting of a node with two children, of which there are 3, and they have 3 even degree nodes, giving 3*3 for a total of 6*1 + 3*3 = 15.
%p a:= n-> n!*coeff(series((T->(T^2+x^2)/
%p (2*T*(1-T)))(-LambertW(-x)), x, n+2), x, n):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jan 03 2015
%Y Cf. A000169, A026641, A000081.
%K nonn
%O 1,2
%A _Marko Riedel_, Jan 03 2015