A124343 Number of rooted trees on n nodes with thinning limbs.

1, 1, 2, 3, 6, 10, 21, 38, 78, 153, 314, 632, 1313, 2700, 5646, 11786, 24831, 52348, 111027, 235834, 502986, 1074739, 2303146, 4944507, 10639201, 22930493, 49511948, 107065966, 231874164, 502834328, 1091842824, 2373565195, 5165713137, 11254029616, 24542260010

Offset

1,3

Comments

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..141

Example

The a(5) = 6 trees are ((((o)))), (o((o))), (o(oo)), ((o)(o)), (oo(o)), (oooo). - Gus Wiseman, Jan 25 2018

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, min(i-1, 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=1..min(k, n-1)))
    end:
a:= n-> A(n$2):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 08 2014

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, Min[i-1, 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, 1, Min[k, n-1] }]];
a[n_] := A[n, n];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000081, A032305, A124344-A124348, A290689, A298303, A298304, A298305, A298422.

Row sums of A244657.

Sequence in context: A178852 A215067 A008928 * A324407 A032291 A063687

Adjacent sequences: A124340 A124341 A124342 * A124344 A124345 A124346

Keyword

nonn

Author

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

Extensions

More terms from Alois P. Heinz, Jul 04 2014

Status

approved