A245120 Number T(n,k) of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=max-index-of-row(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, 1, 0, 1, 8, 2, 0, 1, 12, 4, 0, 1, 22, 9, 0, 1, 36, 17, 2, 0, 1, 63, 35, 3, 0, 1, 107, 67, 9, 0, 1, 188, 131, 20, 0, 1, 327, 249, 46, 1, 0, 1, 578, 484, 94, 4, 0, 1, 1020, 922, 202, 11, 0, 1, 1820, 1775, 412, 28

Offset

1,14

Comments

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.

Links

Alois P. Heinz, Rows n = 1..140, flattened

Example

The A124346(7) = 6 7-node rooted identity 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  : o     o     o       o      : o     :
:  |  : |     |     |              :       :
:  o  : o     o     o              :       :
:  |  : |                          :       :
:  o  : o                          :       :
:  |  :                            :       :
:  o  :                            :       :
:     :                            :       :
: -1- : -------------2------------ : --3-- :
Thus row 7 = [0, 1, 4, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1;
0, 1,  1;
0, 1,  1;
0, 1,  3;
0, 1,  4,  1;
0, 1,  8,  2;
0, 1, 12,  4;
0, 1, 22,  9;
0, 1, 36, 17, 2;
0, 1, 63, 35, 3;

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, add(binomial(A(i, min(i-1, h)), 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=1..min(k, n-1)))
    end:
g:= proc(n) local k; if n=1 then 0 else
       for k while T(n, k)>0 do od; k-1 fi
    end:
T:= (n, k)-> b(n-1$2, k$2):
seq(seq(T(n, k), k=0..g(n)), n=1..25);

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, Sum[Binomial[A[i, Min[i-1, h]], 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, 1, Min[k, n-1]}]]; g[n_] := If[n==1, 0, For[k=1, T[n, k]>0, k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k]; Table[T[n, k], {n, 1, 25}, {k, 0, g[n]}] // Flatten (* Jean-François Alcover, Jan 18 2017, translated from Maple *)

Crossrefs

Column k=0-10 give: A000007(n-1), A000012 (for n>1), A245121, A245122, A245123, A245124, A245125, A245126, A245127, A245128, A245129.

Row sums give A124346.

Cf. A244657.

Sequence in context: A247629 A178116 A238709 * A226912 A177330 A197126

Adjacent sequences: A245117 A245118 A245119 * A245121 A245122 A245123

Keyword

nonn,tabf

Author

Alois P. Heinz, Jul 12 2014

Status

approved