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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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.
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.
Sequence in context: A218321 A236339 A292100 * A059275 A020047 A231496
KEYWORD
nonn
AUTHOR
Stepan Orevkov, Aug 29 2013
EXTENSIONS
Terms a(20) and beyond from Andrew Howroyd, May 21 2018
STATUS
approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)