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

1,2

Links

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

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)

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 *)

Crossrefs

Cf. A316651, A319376.

Sequence in context: A126738 A283298 A259610 * A109074 A357337 A227020

Adjacent sequences: A326393 A326394 A326395 * A326397 A326398 A326399

Keyword

nonn

Author

Alois P. Heinz, Sep 11 2019

Status

approved