A245151 Number T(n,k) of n-node unlabeled rooted trees with thickening limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 5, 1, 0, 0, 1, 0, 7, 3, 1, 0, 0, 1, 0, 12, 3, 1, 0, 0, 0, 1, 0, 17, 8, 1, 1, 0, 0, 0, 1, 0, 28, 9, 3, 1, 0, 0, 0, 0, 1, 0, 42, 21, 3, 1, 1, 0, 0, 0, 0, 1, 0, 69, 28, 5, 1, 1, 0, 0, 0, 0, 0, 1, 0, 105, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Alois P. Heinz, Jul 12 2014

Keywords

• nonn
• tabl

Comments

In a rooted tree with thickening limbs the outdegree of a parent node is smaller than or equal to the outdegree of any of its non-leaf child nodes.

T(n+1,1) = Sum_{k=0..n-1} T(n,k) for n>=1.

T(n+1,n) = T(2n+1,n) = 1 for n>=0.

T(n,1+floor((n-1)/2)) = 0 for n>3.

Examples

			The A245152(5) = 5 5-node rooted trees with thickening limbs sorted by root outdegree are:
: o   o     o   :     o   :    o    :
: |   |     |   :    / \  :  /( )\  :
: o   o     o   :   o   o : o o o o :
: |   |    /|\  :  / \    :         :
: o   o   o o o : o   o   :         :
: |  / \        :         :         :
: o o   o       :         :         :
: |             :         :         :
: o             :         :         :
:               :         :         :
: ------1------ : ---2--- : ---4--- :
Thus row 5 = [0, 3, 1, 0, 1].
Triangle T(n,k) begins:
1;
0,   1;
0,   1,  1;
0,   2,  0,  1;
0,   3,  1,  0, 1;
0,   5,  1,  0, 0, 1;
0,   7,  3,  1, 0, 0, 1;
0,  12,  3,  1, 0, 0, 0, 1;
0,  17,  8,  1, 1, 0, 0, 0, 1;
0,  28,  9,  3, 1, 0, 0, 0, 0, 1;
0,  42, 21,  3, 1, 1, 0, 0, 0, 0, 1;
0,  69, 28,  5, 1, 1, 0, 0, 0, 0, 0, 1;
0, 105, 56,  9, 3, 1, 1, 0, 0, 0, 0, 0, 1;
0, 176, 81, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Columns k=0-10 give: A000007(n-1), A245152(n-1), A245142, A245143, A245144, A245145, A245146, A245147, A245148, A245149, A245150.

Row sums give A245152.

Cf. A244657 (thinning limbs).

Programs

Maple

b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, k$2):
seq(seq(T(n, k), k=0..n-1), n=1..20);

Mathematica

b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i<1 || v<1 || nJean-François Alcover, Jan 27 2015, after Alois P. Heinz *)