OFFSET
1,4
COMMENTS
There is no planted tree on one node by definition.
Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012
It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
FORMULA
Shifts left 2 places under Euler transform.
G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014
G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...). - Ilya Gutkovskiy, Jun 11 2021
EXAMPLE
G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ...
From Joerg Arndt, Jun 23 2014: (Start)
The a(8) = 20 such trees have the following level sequences:
01: [ 0 1 2 3 4 3 2 1 ]
02: [ 0 1 2 3 3 3 2 1 ]
03: [ 0 1 2 3 3 2 2 1 ]
04: [ 0 1 2 3 3 2 1 1 ]
05: [ 0 1 2 3 2 3 2 1 ]
06: [ 0 1 2 3 2 2 2 1 ]
07: [ 0 1 2 3 2 2 1 1 ]
08: [ 0 1 2 3 2 1 2 1 ]
09: [ 0 1 2 3 2 1 1 1 ]
10: [ 0 1 2 2 2 2 2 1 ]
11: [ 0 1 2 2 2 2 1 1 ]
12: [ 0 1 2 2 2 1 2 1 ]
13: [ 0 1 2 2 2 1 1 1 ]
14: [ 0 1 2 2 1 2 2 1 ]
15: [ 0 1 2 2 1 2 1 1 ]
16: [ 0 1 2 2 1 1 1 1 ]
17: [ 0 1 2 1 2 1 2 1 ]
18: [ 0 1 2 1 2 1 1 1 ]
19: [ 0 1 2 1 1 1 1 1 ]
20: [ 0 1 1 1 1 1 1 1 ]
Successive levels change by at most 1 and the last level is 1, compare to the example in A000081.
(End)
From Gus Wiseman, Aug 22 2018: (Start)
The a(7) = 10 locally non-intersecting trees:
(o(o(oo)))
(o(oo(o)))
(o(oooo))
(oo(o(o)))
(oo(ooo))
(o(o)(oo))
(ooo(oo))
(oo(o)(o))
(oooo(o))
(oooooo)
(End)
MAPLE
with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40); # Alois P. Heinz, Sep 06 2008
MATHEMATICA
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)
purt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]], Intersection@@#=={}&], {ptn, IntegerPartitions[n-1]}]];
Table[Length[purt[n]], {n, 10}] (* Gus Wiseman, Aug 22 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */
CROSSREFS
KEYWORD
nonn,nice,eigen
AUTHOR
EXTENSIONS
Better description from Christian G. Bower, May 15 1998
STATUS
approved