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

0, 0, 1, 2, 5, 10, 23, 52, 120, 275, 644, 1508, 3558, 8418, 20012, 47699, 114082, 273476, 657250, 1582817, 3819514, 9233059, 22356918, 54216429, 131663670, 320158789, 779461271, 1899830067, 4635492672, 11321595218, 27677333555, 67720658475, 165835173692

Offset: 0

Author

Alois P. Heinz, Aug 26 2017

Keywords

• nonn

Examples

			: a(5) = 10 = 4 + 3 + 3 : a(4) = 5 = 3 + 2 :
:                       :                  :
:    o    o      o      :    o    o        :
:    |    |     / \     :    |   / \       :
:    o    o    o   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..300

Crossrefs

Cf. A001853, A004111, A291529.

Programs

Maple

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
      `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
a:= n-> add(add((h+1)*F(n-1, h, t), t=1..n-1-h), h=0..n-2):
seq(a(n), n=0..37);

Mathematica

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1,
   If[n < 2, x^(t*n), 0], b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
   b[i-1, i-1, 0, h-1], j]*b[n-i*j, i-1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
a[n_] := Sum[Sum[(h+1)*F[n-1, h, t], {t, 1, n-1-h}], {h, 0, n-2}];
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * A291529(n-1,h,t).