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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Index entries for sequences related to rooted trees

Example

: 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
:

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<v, 0,
      `if`(v=n, 1, add(binomial(A(i, k)+j-1, j)*
       b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
    end:
A:= proc(n, k) option remember; `if`(n<2, n,
      add(b(n, n+1-j, j, k), j=2..min(n, k)))
    end:
a:= 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 *)

Crossrefs

Limit of reversed rows of A292086.

Sequence in context: A292234 A292235 A292236 * A091702 A274104 A068593

Adjacent sequences: A292084 A292085 A292086 * A292088 A292089 A292090

Keyword

nonn

Author

Alois P. Heinz, Sep 08 2017

Status

approved