A319580 Number of binary rooted trees with n leaves of n colors and all non-leaf nodes having out-degree 2.

1, 3, 18, 215, 3600, 80136, 2213036, 73068543, 2806959015, 123002168300, 6055381161852, 330885794632536, 19872950226273053, 1301261803764756855, 92259974680854975000, 7041606755629152575055, 575638367425376279620662, 50180725346542105445190603

Offset: 1

Author

Andrew Howroyd, Sep 23 2018

Keywords

• nonn

Comments

Not all of the n colors need to be used.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..352
• V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Crossrefs

Main diagonal of A319539.

Cf. A319369, A319541.

Programs

Maple

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
    end:
a:= n-> A(n$2):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 23 2018

Mathematica

A[n_, k_] := A[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1-t) / 2][A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]];
a[n_] := A[n, n];
Array[a, 20] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)

PARI

a(n)={my(v=vector(n)); v[1]=n; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v[n]} \\ Andrew Howroyd, Sep 23 2018