A319369 Number of series-reduced rooted trees with n leaves of n colors.

1, 3, 28, 430, 9376, 269675, 9632960, 411395268, 20445999734, 1159248404721, 73846864163348, 5221802726902476, 405858598184643930, 34392275731729465799, 3155760058245300968416, 311720334688779807141832, 32980137195294216968253900, 3720954854814866649904474180

Offset: 1

Author

Andrew Howroyd, Sep 17 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..340
• V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Crossrefs

Main diagonal of A319254.

Cf. A000311 (1 leaf of each color), A316651.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> A(n$2):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := A[n, n];
a /@ Range[1, 20] (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
a(n)={my(v=[n]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v[n]}

Formula

a(n) ~ c * d^n * n^(n - 3/2), where d = 1/(2*log(2) - 1) = 2.588699449562089830805384431942090... and c = 0.2580000331300831455241033648... - Vaclav Kotesovec, Sep 18 2019, updated Mar 16 2024