A244710 -id:A244710 - cp's OEIS frontend

A244657 Number T(n,k) of n-node unlabeled rooted trees with thinning 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, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 9, 6, 3, 1, 1, 0, 1, 13, 13, 6, 3, 1, 1, 0, 1, 26, 25, 15, 6, 3, 1, 1, 0, 1, 42, 55, 29, 15, 6, 3, 1, 1, 0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1, 0, 1, 138, 224, 140, 72, 31, 15, 6, 3, 1, 1

Offset: 1

Alois P. Heinz, Jul 04 2014

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

			The A124343(5) = 6 5-node rooted trees with thinning 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   o : o     :         :
:  |  : |                   :       :         :
:  o  : o                   :       :         :
:  |  :                     :       :         :
:  o  :                     :       :         :
:     :                     :       :         :
: -1- : ---------2--------- : --3-- : ---4--- :
Thus row 5 = [0, 1, 3, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,   1;
  0, 1,  3,   1,  1;
  0, 1,  4,   3,  1,  1;
  0, 1,  9,   6,  3,  1,  1;
  0, 1, 13,  13,  6,  3,  1, 1;
  0, 1, 26,  25, 15,  6,  3, 1, 1;
  0, 1, 42,  55, 29, 15,  6, 3, 1, 1;
  0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1;

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

Columns k=0-10 give: A000007(n-1), A000012 (for n>1), A244703, A244704, A244705, A244706, A244707, A244708, A244709, A244710, A244711.
T(2n,n) gives A244712.
Row sums give A124343.
Cf. A245120, A245151.

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..14);

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, Feb 03 2015, after Alois P. Heinz *)

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A244657 Number T(n,k) of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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