A102755 Number of asymmetric (or identity) oriented trees with n nodes.

1, 1, 1, 4, 10, 37, 135, 522, 2060, 8430, 35115, 149286, 644456, 2821835, 12503878, 56001856, 253174451, 1154179790, 5301178673, 24513058220, 114042743290, 533510321377, 2508491383101, 11849321038092, 56211286929146, 267707017974770, 1279602152054934

Offset: 1

Author

Vladeta Jovovic, Feb 10 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A005753 = number of asymmetric (or identity) rooted oriented trees with n nodes.

Cf. A246312.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*b(i-1$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n-1$2)-add(b(j-1$2)*b(n-j-1$2), j=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[2*b[i-1, i-1], j]*b[n - i*j, i-1], {j, 0, n/i}]]] // FullSimplify; a[n_] := b[n-1, n-1] - Sum[b[j-1, j-1]*b[n-j-1, n-j-1], {j, 1, n-1}]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

G.f.: B(x)-B(x)^2, where B(x) is g.f. for A005753.

a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.17807103914078424643862998... . - Vaclav Kotesovec, Aug 25 2014