A038075 Number of rooted identity trees with 2-colored leaves.

2, 2, 3, 7, 16, 41, 110, 304, 858, 2474, 7234, 21418, 64057, 193277, 587531, 1797817, 5532916, 17115442, 53186682, 165958893, 519764706, 1633331926, 5148420607, 16273962742, 51574291758, 163834983761, 521597902077, 1663993969029, 5318540288800, 17029516243797

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, A038076.

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, 2*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, 2*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 = 3.3683668081969694736300401764..., c = 0.4229796097587478606873477... . - Vaclav Kotesovec, Sep 10 2014

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