A038076 Number of rooted identity trees with 3-colored leaves.

3, 3, 6, 16, 46, 142, 461, 1542, 5278, 18417, 65218, 233816, 846938, 3094943, 11395715, 42237936, 157465847, 590075550, 2221391912, 8397223487, 31861406058, 121300625969, 463233477550, 1774034788166, 6811612470692, 26216538077715, 101125406981562

Offset: 1

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..600
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A004111, A038075.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, 3*n, b(n-1, n-1)):
seq(a(n), n=1..35);  # Alois P. Heinz, Aug 01 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<2, 3*n, b[n-1, n-1]];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

Shifts left under Weigh transform.

a(n) ~ c * d^n / n^(3/2), where d = 4.0814589930714884560076189705..., c = 0.4583632659157592121544633778... . - Vaclav Kotesovec, Sep 06 2014

G.f. A(x) satisfies: A(x) = 2*x + x * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k) / k ). - Ilya Gutkovskiy, May 19 2023