A339644 Number of rooted trees on n nodes with labels covering an initial interval of positive integers.

1, 3, 21, 214, 3004, 53696, 1169220, 30017582, 887835091, 29728120594, 1111619802614, 45914106227815, 2076062017348677, 101996651482313080, 5410363994433018486, 308174409706787225523, 18760485689929220881741, 1215547422537201878074293, 83520534389622385511232635

Offset: 1

Author

Andrew Howroyd, Dec 11 2020

Keywords

• nonn

Examples

			The a(3) = 21 rooted trees are:
  (1(11)), (1(1(1))), (1(12)), (1(22)), (1(1(2))), (1(2(1))), (1(2(2))), (2(12)), (2(11)), (2(2(1))), (2(1(2))), (2(1(1))), (1(23)), (1(2(3))), (1(3(2))), (2(13)), (2(1(3))), (2(3(1))), (3(12)), (3(1(2))), (3(2(1))).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Row sums of A141610.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
      d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
    end:
a:= n-> add(add(b(n, k-j)*binomial(k, j)*(-1)^j, j=0..k), k=0..n):
seq(a(n), n=1..21);  # Alois P. Heinz, Dec 11 2020

Mathematica

b[n_, k_] := b[n, k] = If[n<2, k*n, (Sum[Sum[b[d, k]*d, {d, Divisors[j]}]* b[n - j, k], {j, 1, n - 1}])/(n - 1)];
a[n_] := Sum[Sum[b[n, k - j]*Binomial[k, j]*(-1)^j, {j, 0, k}], {k, 0, n}];
Array[a, 21] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)

PARI

\\ See A141610 for U(n, k).
seq(n)={sum(k=1, n, U(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}