cp's OEIS Frontend

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

%I A291532 #16 Apr 29 2022 13:36:45
%S A291532 0,1,1,3,4,9,19,40,84,186,413,922,2082,4733,10831,24928,57648,133923,
%T A291532 312393,731328,1717784,4047111,9561517,22647521,53770164,127941813,
%U A291532 305046676,728688803,1743752229,4179697971,10034077377,24123567285,58076419495,139996849639
%N A291532 Number of trees in all forests of (unlabeled) rooted identity trees with n vertices.
%H A291532 Alois P. Heinz, <a href="/A291532/b291532.txt">Table of n, a(n) for n = 0..1000</a>
%F A291532 a(n) = Sum_{k>=1} k * A227774(n+1,k).
%F A291532 a(n) = Sum_{h=0..n} Sum_{t=0..n-h} t * A291529(n,h,t).
%e A291532 a(4) = 4:
%e A291532 :   o   :   o  o   :     o     :
%e A291532 :   |   :   |      :    / \    :
%e A291532 :   o   :   o      :   o   o   :
%e A291532 :   |   :   |      :   |       :
%e A291532 :   o   :   o      :   o       :
%e A291532 :   |   :          :           :
%e A291532 :   o   :          :           :
%e A291532 :       :          :           :
%p A291532 b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
%p A291532        `if`(n<2, x^(t*n), 0), b(n, i-1, t)+add(binomial(
%p A291532        b(i-1$2, 0), j)*x^(t*j)*b(n-i*j, i-1, t), j=1..n/i)))
%p A291532     end:
%p A291532 a:= n-> (p-> add(i*coeff(p,x,i), i=1..degree(p)))(b(n$2, 1)):
%p A291532 seq(a(n), n=0..35);
%t A291532 b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0 || i == 1,
%t A291532      If[n < 2, x^(t*n), 0], b[n, i - 1, t] + Sum[Binomial[
%t A291532      b[i - 1, i - 1, 0], j]*x^(t*j)*b[n - i*j, i - 1, t], {j, 1, n/i}]]];
%t A291532 a[n_] := Function[p, Sum[i*Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][
%t A291532      b[n, n, 1]];
%t A291532 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Apr 29 2022, after _Alois P. Heinz_ *)
%Y A291532 Cf. A004111, A005197, A227774, A291529.
%K A291532 nonn
%O A291532 0,4
%A A291532 _Alois P. Heinz_, Aug 25 2017