A299113 Number of rooted identity trees with 2n+1 nodes.

1, 1, 3, 12, 52, 247, 1226, 6299, 33209, 178618, 976296, 5407384, 30283120, 171196956, 975662480, 5599508648, 32334837886, 187737500013, 1095295264857, 6417886638389, 37752602033079, 222861754454841, 1319834477009635, 7839314017612273, 46688045740233741

Offset: 0

Author

Alois P. Heinz, Feb 02 2018

Keywords

• nonn

Examples

			a(2) = 3:
   o     o       o
   |     |      / \
   o     o     o   o
   |    / \    |
   o   o   o   o
   |   |       |
   o   o       o
   |
   o
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1253

Crossrefs

Bisection of A004111 (odd part).

Cf. A100427, A299098.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(
      b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
    end:
a:= n-> b(2*n+1):
seq(a(n), n=0..30);

Mathematica

b[n_] := b[n] = If[n < 2, n, Sum[b[n - k]*Sum[b[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
a[n_] := b[2*n + 1];
Array[a, 30, 0] (* Jean-François Alcover, May 30 2019, from Maple *)

Python

from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def b(n): return n if n<2 else sum([b(n-k)*sum([b(d)*d*(-1)**(k//d+1) for d in divisors(k)]) for k in range(1, n)])//(n-1)
def a(n): return b(2*n+1)
print([a(n) for n in range(31)]) # Indranil Ghosh, Mar 02 2018

Formula

a(n) = A004111(2n+1).