A117356 Number of rooted trees with total weight n, where the weight of a node at height k is k (with the root considered to be at level 0).

1, 1, 1, 2, 2, 3, 5, 6, 8, 12, 16, 22, 31, 41, 56, 78, 104, 142, 194, 260, 353, 478, 641, 864, 1164, 1560, 2095, 2810, 3757, 5028, 6722, 8966, 11963, 15945, 21223, 28244, 37551, 49871, 66210, 87829, 116411, 154222, 204162, 270084, 357117, 471881, 623146

Offset: 0

Author

Franklin T. Adams-Watters, Mar 09 2006

Keywords

• nonn

Comments

Equivalently, number of forests of total weight n, when the roots are considered to be at height 1; so this is the Euler transform of A117357. - Franklin T. Adams-Watters, Oct 03 2009

Examples

			a(3) = 2; there is one tree with 3 nodes at height 1 and one with 1 node at height 1 and 1 at height 2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000081, A115593, A117357.

Programs

Maple

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i g(n, n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, May 16 2013

Mathematica

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < k, 0, Sum[Binomial[g[i - k, i - k, k + 1] + j - 1, j] g[n - i j, i - 1, k], {j, 0, n/i}]]];
a[n_] := g[n, n, 1];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

Formula

If a(n) is the equivalent of this sequence with the root node considered to be at level k, then a(n) is the Euler transform of a(n) shifted right k places. To compute N terms, take k so that (k+1)*(k+2)/2 > N, approximate a(n) by 1 if n=k, 0 otherwise and apply this rule repeatedly. Formula from Christian G. Bower.