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%I #14 Jun 07 2021 12:35:22
%S 0,1,2,6,28,200,1926,22512,306104,4770432,84234250,1663735040,
%T 36320155092,867963393024,22535294920334,631718010255360,
%U 19016907901995376,611869203759792128,20954324710009221138,761015341362413371392,29214930870257449355660
%N Number of labeled rooted trees on n nodes such that each internal node has an odd number of children.
%H Alois P. Heinz, <a href="/A216187/b216187.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f. satisfies: F(x) = x*(sinh(F(x))+1).
%F a(n) ~ sqrt(s/(s-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.482309923717218507261475229723265094762759829863... and s = 1.358310572965774067065006624540704170183889018218... are real roots of the system of equations s = r*(1 + sinh(s)), r*cosh(s) = 1. - _Vaclav Kotesovec_, Jun 07 2021
%e a(5) = 200: There are three unlabeled rooted trees of 5 nodes with all internal nodes having an odd number of children. They can be labeled respectively in 20 + 120 + 60 = 200 ways.
%e ..o............o............o....
%e ..|............|.........../|\...
%e ..o............o..........o.o.o..
%e ./|\...........|..........|......
%e o.o.o..........o..........o......
%e ...............|.................
%e ...............o.................
%e ...............|.................
%e ...............o.................
%p a:= n-> n!*coeff(series(RootOf(F=x*(sinh(F)+1), F), x, n+1), x, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 12 2013
%t nn=12; f[x_]:=Sum[a[n]x^n/n!, {n,0,nn}]; s=SolveAlways[0==Series[f[x]-x (Sinh[f[x]]+1), {x,0,nn}], x]; Table[a[n], {n,0,nn}]/.s
%Y Cf. A036778.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Mar 11 2013
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