cp's OEIS Frontend

A291559 Total height of all (unlabeled) rooted identity trees with n vertices.

%I A291559 #14 Dec 08 2023 07:12:15
%S A291559 0,0,1,2,5,10,23,52,120,275,644,1508,3558,8418,20012,47699,114082,
%T A291559 273476,657250,1582817,3819514,9233059,22356918,54216429,131663670,
%U A291559 320158789,779461271,1899830067,4635492672,11321595218,27677333555,67720658475,165835173692
%N A291559 Total height of all (unlabeled) rooted identity trees with n vertices.
%H A291559 Alois P. Heinz, <a href="/A291559/b291559.txt">Table of n, a(n) for n = 0..300</a>
%F A291559 a(n) = Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * A291529(n-1,h,t).
%e A291559 : a(5) = 10 = 4 + 3 + 3 : a(4) = 5 = 3 + 2 :
%e A291559 :                       :                  :
%e A291559 :    o    o      o      :    o    o        :
%e A291559 :    |    |     / \     :    |   / \       :
%e A291559 :    o    o    o   o    :    o  o   o      :
%e A291559 :    |   / \   |        :    |  |          :
%e A291559 :    o  o   o  o        :    o  o          :
%e A291559 :    |  |      |        :    |             :
%e A291559 :    o  o      o        :    o             :
%e A291559 :    |                  :                  :
%e A291559 :    o                  :                  :
%e A291559 :                       :                  :
%p A291559 b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
%p A291559       `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
%p A291559        b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
%p A291559     end:
%p A291559 g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
%p A291559 F:= (n, h, t)-> coeff(g(n, h), x, t):
%p A291559 a:= n-> add(add((h+1)*F(n-1, h, t), t=1..n-1-h), h=0..n-2):
%p A291559 seq(a(n), n=0..37);
%t A291559 b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1,
%t A291559    If[n < 2, x^(t*n), 0], b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
%t A291559    b[i-1, i-1, 0, h-1], j]*b[n-i*j, i-1, t, h], {j, 1, n/i}]]];
%t A291559 g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
%t A291559 F[n_, h_, t_] := Coefficient[g[n, h], x, t];
%t A291559 a[n_] := Sum[Sum[(h+1)*F[n-1, h, t], {t, 1, n-1-h}], {h, 0, n-2}];
%t A291559 Table[a[n], {n, 0, 37}] (* _Jean-François Alcover_, Dec 08 2023, after _Alois P. Heinz_ *)
%Y A291559 Cf. A001853, A004111, A291529.
%K A291559 nonn
%O A291559 0,4
%A A291559 _Alois P. Heinz_, Aug 26 2017