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%I #31 Nov 28 2022 09:59:56
%S 1,2,8,39,214,1268,7949,51901,349703,2415348,17020341,121939535,
%T 885841162,6511874216,48359860685,362343773669,2736184763500,
%U 20805175635077,159174733727167,1224557214545788,9467861087020239,73534456468877012,573484090227222260
%N a(n) is the number of rooted trees with 2n vertices n of whom are leaves.
%H Andrew Howroyd, <a href="/A185650/b185650.txt">Table of n, a(n) for n = 1..200</a>
%H V. M. Kharlamov and S. Yu. Orevkov, <a href="http://arxiv.org/abs/math/0301245">The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves</a>, arXiv:math/0301245 [math.AG], 2003; J. of Combinatorial Theory, Ser. A, 105 (2004), 127-142.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e From _Gus Wiseman_, Nov 27 2022: (Start)
%e The a(1) = 1 through a(3) = 8 rooted trees:
%e (o) ((oo)) (((ooo)))
%e (o(o)) ((o)(oo))
%e ((o(oo)))
%e ((oo(o)))
%e (o((oo)))
%e (o(o)(o))
%e (o(o(o)))
%e (oo((o)))
%e (End)
%t terms = 23;
%t m = 2 terms;
%t T[_, _] = 0;
%t Do[T[x_, z_] = z x - x + x Exp[Sum[Series[1/k T[x^k, z^k], {x, 0, j}, {z, 0, j}], {k, 1, j}]] // Normal, {j, 1, m}];
%t cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
%t Table[cc[[2n+1, n+1]], {n, 1, terms}] (* _Jean-François Alcover_, Sep 14 2018 *)
%t art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[Select[art[n],Count[#,{},{-2}]==n/2&]],{n,2,10,2}] (* _Gus Wiseman_, Nov 27 2022 *)
%o (PARI) \\ here R is A055277 as vector of polynomials
%o R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1 + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
%o {my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k],k))} \\ _Andrew Howroyd_, May 21 2018
%Y The ordered version is A000891, ranked by A358579.
%Y This is the central column of A055277.
%Y These trees are ranked by A358578.
%Y For height = internals we have A358587.
%Y Square trees are counted by A358589.
%Y A000081 counts rooted trees, ordered A000108.
%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358575 counts rooted trees by nodes and internals, ordered A090181.
%Y Cf. A034781, A109129, A358580, A358581-A358584, A358591.
%K nonn
%O 1,2
%A _Stepan Orevkov_, Aug 29 2013
%E Terms a(20) and beyond from _Andrew Howroyd_, May 21 2018
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