A299039 Number of rooted trees with 2n nodes where each node has at most n children.

1, 1, 3, 17, 106, 693, 4690, 32754, 234746, 1719325, 12820920, 97039824, 743680508, 5759507657, 45006692668, 354425763797, 2809931206626, 22409524536076, 179655903886571, 1447023307374888, 11703779855021636, 95020085240320710, 774088021528328920

Offset: 0

Author

Alois P. Heinz, Feb 01 2018

Keywords

• nonn

Examples

			a(2) = 3:
   o     o       o
   |     |      / \
   o     o     o   o
   |    / \    |
   o   o   o   o
   |
   o
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..275

Crossrefs

Cf. A051491, A100034, A187770, A244407, A299038, A299098.

Programs

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n-1$2, n$2)):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[2n - 1, 2n - 1, n, n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

a(n) = A299038(2n,n).

a(n) ~ c * d^n / n^(3/2), where d = A051491^2 = 8.736548423865419449938118272879... and c = A187770 / 2^(3/2) = 0.155536626247883986039760097126... - Vaclav Kotesovec, Feb 02 2018, updated Mar 17 2024