A319220 Number of rooted identity trees with n colored non-root nodes where the set of colors equals {1,...,k} for some k <= n.

1, 1, 4, 32, 362, 5454, 102469, 2312418, 60994931, 1842667249, 62760237328, 2379922607427, 99460696044565, 4542324964768755, 225087388544097949, 12029089158757401655, 689679033455762592599, 42228989406791157626917, 2750301966874829159250696

Offset: 0

Author

Alois P. Heinz, Sep 13 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..368

Crossrefs

Cf. A256068.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, n, add(b(n-j, k)*add(b(d, k)
      *k*d*(-1)^(j/d+1), d=numtheory[divisors](j)), j=1..n-1)/(n-1))
    end:
a:= n-> add(add(b(n+1, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..20);

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, n, Sum[b[n - j, k]*Sum[b[d, k]*
     k*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n - 1}]/(n - 1)];
a[n_] := Sum[Sum[b[n+1, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 02 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} A256068(n+1,k).