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

1,8

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.

Links

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

Example

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;

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<v, 0,
      `if`(n=v, 1, add(binomial(A(i, h)+j-1, j)*
       b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
    end:
A:= proc(n, k) option remember;
      `if`(n<2, n, add(b(n-1$2, j$2), j=k..n-1))
    end:
T:= (n, k)-> 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 || n<v, 0, If[n == v, 1, Sum[Binomial[A[i, h] + j - 1, j]*b[n - i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]]; A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, k, n-1}]]; T[n_, k_] := b[n-1, n-1, k, k]; Table[ Table[T[n, k], {k, 0, n - 1}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

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).

Sequence in context: A318371 A363157 A363952 * A243978 A356898 A106844

Adjacent sequences: A245148 A245149 A245150 * A245152 A245153 A245154

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 12 2014

Status

approved