

A185650


a(n) is the number of rooted trees with 2n vertices n of whom are leaves.


31



1, 2, 8, 39, 214, 1268, 7949, 51901, 349703, 2415348, 17020341, 121939535, 885841162, 6511874216, 48359860685, 362343773669, 2736184763500, 20805175635077, 159174733727167, 1224557214545788, 9467861087020239, 73534456468877012, 573484090227222260
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OFFSET

1,2


LINKS



EXAMPLE

The a(1) = 1 through a(3) = 8 rooted trees:
(o) ((oo)) (((ooo)))
(o(o)) ((o)(oo))
((o(oo)))
((oo(o)))
(o((oo)))
(o(o)(o))
(o(o(o)))
(oo((o)))
(End)


MATHEMATICA

terms = 23;
m = 2 terms;
T[_, _] = 0;
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}];
cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n1]}]];
Table[Length[Select[art[n], Count[#, {}, {2}]==n/2&]], {n, 2, 10, 2}] (* Gus Wiseman, Nov 27 2022 *)


PROG

(PARI) \\ here R is A055277 as vector of polynomials
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)};
{my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k], k))} \\ Andrew Howroyd, May 21 2018


CROSSREFS

This is the central column of A055277.
For height = internals we have A358587.
Square trees are counted by A358589.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



