A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

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

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

V. M. Kharlamov and S. Yu. Orevkov, The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, arXiv:math/0301245 [math.AG], 2003; J. of Combinatorial Theory, Ser. A, 105 (2004), 127-142.

Index entries for sequences related to rooted trees

Example

From Gus Wiseman, Nov 27 2022: (Start)
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];
Table[cc[[2n+1, n+1]], {n, 1, terms}] (* Jean-François Alcover, Sep 14 2018 *)
art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
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

The ordered version is A000891, ranked by A358579.

This is the central column of A055277.

These trees are ranked by A358578.

For height = internals we have A358587.

Square trees are counted by A358589.

A000081 counts rooted trees, ordered A000108.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

A358575 counts rooted trees by nodes and internals, ordered A090181.

Cf. A034781, A109129, A358580, A358581-A358584, A358591.

Sequence in context: A218321 A236339 A292100 * A059275 A020047 A231496

Adjacent sequences: A185647 A185648 A185649 * A185651 A185652 A185653

Keyword

nonn

Author

Stepan Orevkov, Aug 29 2013

Extensions

Terms a(20) and beyond from Andrew Howroyd, May 21 2018

Status

approved