A292087 Limit of the number of (unlabeled) rooted trees without unary nodes where n is the difference between the number of leafs and the maximal outdegree as the tree size increases.

1, 2, 7, 23, 78, 262, 893, 3040, 10411, 35724, 122950, 424004, 1465254, 5071981, 17584226, 61046464, 212197118, 738422362, 2572261241, 8968726829, 31298189180, 109307655964, 382031357974, 1336107044159, 4675807680776, 16372936282017, 57363325974309

Offset: 0

Author

Alois P. Heinz, Sep 08 2017

Keywords

• nonn

Examples

			: a(0) = 1:
:                  o
:               //( )\\
:             N N N N N N
:
: a(1) = 2:
:             o               o
:           /   \          / /|\ \
:          o     N        o N N N N
:       / /|\ \          ( )
:      N N N N N         N N
:
: a(2) = 7:
:           o             o            o             o
:          / \          /   \        /( )\         / | \
:         o   N        o     N      o N N N       o  N  N
:        / \         /( )\         / \          /( )\
:       o   N       o N N N       o   N        N N N N
:     /( )\        ( )           ( )
:    N N N N       N N           N N
:
:          o            o              o
:        /   \        /( )\         / ( \ \
:       o     o      o N N N      o   o  N N
:     /( )\  ( )    /|\          ( ) ( )
:    N N N N N N   N N N         N N N N
:
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Index entries for sequences related to rooted trees

Crossrefs

Limit of reversed rows of A292086.

Programs

Maple

b:= proc(n, i, v, k) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n A(2*n+3, n+3)-A(2*n+3, n+2):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0,
    If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
    If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*
    b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n,
    Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
a[n_] := A[2*n + 3, n + 3] - A[2*n + 3, n + 2];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 28 2024, after Alois P. Heinz *)