A309994 Number of forests of rooted trees with 2n colored nodes using exactly n colors.

1, 2, 89, 14845, 5613775, 3809941836, 4073969863427, 6316651717425358, 13407079935176225215, 37344967651943608528498, 132181958309965092862822183, 579566807739313784043087337938, 3083812115454145185391757131500066, 19577110356940490275990571617295644659

Offset: 0

Author

Alois P. Heinz, Aug 26 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..187

Crossrefs

Cf. A256064.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*b(d, k),
      d=numtheory[divisors](j))*b(n-j, k)*k, j=1..n-1))/(n-1))
    end:
a:= n-> add(b(2*n+1, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..15);

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, n, Sum[Sum[d*b[d, k], {d, Divisors[j]}]*b[n-j, k]*k, {j, 1, n-1}]/(n-1)];
a[n_] := Sum[b[2*n+1, n-i]*(-1)^i*Binomial[n, i], {i, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 15 2022, after Alois P. Heinz *)

Formula

a(n) = A256064(2*n+1,n).