A242375 Number of rooted trees with n n-colored non-root nodes.

1, 1, 7, 82, 1499, 37476, 1200705, 46990952, 2175619923, 116400215521, 7069820334023, 480722969498938, 36186340018129392, 2987845924408179654, 268530017303221572650, 26098422892000807053155, 2727654868575748827350403, 305075571192329680642519141

Offset: 0

Author

Alois P. Heinz, May 12 2014

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

A diagonal of A242249.

Cf. A255523.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      b(d, k), d=divisors(j))*b(n-j, k)*k, j=1..n-1))/(n-1))
    end:
a:= n-> b(n+1, n):
seq(a(n), n=0..20);

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, n, (Sum[Sum[d*b[d, k], {d, Divisors[j]}] * b[n - j, k]*k, {j, 1, n - 1}])/(n - 1)];
a[n_] := b[n + 1, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

Formula

a(n) ~ c * exp(n) * n^(n-3/2), where c = exp(1 + exp(-2)/2) / sqrt(2*Pi) = 1.160358615244339554387715748... . - Vaclav Kotesovec, Aug 28 2014, updated Mar 18 2024