A038077 Number of rooted identity trees with 2-colored nodes.

2, 4, 10, 36, 132, 532, 2222, 9584, 42248, 189776, 864830, 3989656, 18593424, 87413444, 414060796, 1974260912, 9467922870, 45638482068, 221001289714, 1074591477112, 5244497440468, 25681907242416, 126149132242490, 621386729646752, 3068748094479108

Offset: 1

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn
• eigen

Comments

Shifts left and halves under Weigh transform.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A004111, A038078, A038079, A038080, A246312.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 2, 2*b((n-1)$2)):
seq(a(n), n=1..40); # Alois P. Heinz, May 20 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==1, 2, 2*b[n-1, n-1]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

a(n) = 2 * A005753(n).