A326396 Total number of colors in all series-reduced rooted trees with n leaves where colors span an initial interval of the color palette.

1, 3, 26, 322, 5210, 104421, 2491498, 68907073, 2166242180, 76266794945, 2972079029674, 126987589678185, 5902427979920102, 296484317531254557, 16003975713659818226, 923838934059255332723, 56788871072327503930862, 3703444074072753204057172

Offset: 1

Author

Alois P. Heinz, Sep 11 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A316651, A319376.

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-> add(k*add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):
seq(a(n), n=1..20);

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_] := Sum[k Sum[A[n, k-j](-1)^j Binomial[k, j], {j, 0, k-1}], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A319376(n,k).

From Vaclav Kotesovec, Sep 18 2019: (Start)

a(n) ~ c * d^n * n^n, where d = 1.37392076830840090205551979... and c = 0.29889555940946459367729...

a(n) ~ n*A316651(n)/(2*log(2)). (End)