A291532 Number of trees in all forests of (unlabeled) rooted identity trees with n vertices.

0, 1, 1, 3, 4, 9, 19, 40, 84, 186, 413, 922, 2082, 4733, 10831, 24928, 57648, 133923, 312393, 731328, 1717784, 4047111, 9561517, 22647521, 53770164, 127941813, 305046676, 728688803, 1743752229, 4179697971, 10034077377, 24123567285, 58076419495, 139996849639

Offset: 0

Author

Alois P. Heinz, Aug 25 2017

Keywords

• nonn

Examples

			a(4) = 4:
:   o   :   o  o   :     o     :
:   |   :   |      :    / \    :
:   o   :   o      :   o   o   :
:   |   :   |      :   |       :
:   o   :   o      :   o       :
:   |   :          :           :
:   o   :          :           :
:       :          :           :
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A004111, A005197, A227774, A291529.

Programs

Maple

b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
       `if`(n<2, x^(t*n), 0), b(n, i-1, t)+add(binomial(
       b(i-1$2, 0), j)*x^(t*j)*b(n-i*j, i-1, t), j=1..n/i)))
    end:
a:= n-> (p-> add(i*coeff(p,x,i), i=1..degree(p)))(b(n$2, 1)):
seq(a(n), n=0..35);

Mathematica

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0 || i == 1,
     If[n < 2, x^(t*n), 0], b[n, i - 1, t] + Sum[Binomial[
     b[i - 1, i - 1, 0], j]*x^(t*j)*b[n - i*j, i - 1, t], {j, 1, n/i}]]];
a[n_] := Function[p, Sum[i*Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][
     b[n, n, 1]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=1} k * A227774(n+1,k).

a(n) = Sum_{h=0..n} Sum_{t=0..n-h} t * A291529(n,h,t).