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

%I A339644 #14 Jan 04 2021 09:56:52
%S A339644 1,3,21,214,3004,53696,1169220,30017582,887835091,29728120594,
%T A339644 1111619802614,45914106227815,2076062017348677,101996651482313080,
%U A339644 5410363994433018486,308174409706787225523,18760485689929220881741,1215547422537201878074293,83520534389622385511232635
%N A339644 Number of rooted trees on n nodes with labels covering an initial interval of positive integers.
%H A339644 Andrew Howroyd, <a href="/A339644/b339644.txt">Table of n, a(n) for n = 1..200</a>
%e A339644 The a(3) = 21 rooted trees are:
%e A339644   (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))).
%p A339644 b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
%p A339644       d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
%p A339644     end:
%p A339644 a:= n-> add(add(b(n, k-j)*binomial(k, j)*(-1)^j, j=0..k), k=0..n):
%p A339644 seq(a(n), n=1..21);  # _Alois P. Heinz_, Dec 11 2020
%t A339644 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)];
%t A339644 a[n_] := Sum[Sum[b[n, k - j]*Binomial[k, j]*(-1)^j, {j, 0, k}], {k, 0, n}];
%t A339644 Array[a, 21] (* _Jean-François Alcover_, Jan 04 2021, after _Alois P. Heinz_ *)
%o A339644 (PARI) \\ See A141610 for U(n, k).
%o A339644 seq(n)={sum(k=1, n, U(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}
%Y A339644 Row sums of A141610.
%K A339644 nonn
%O A339644 1,2
%A A339644 _Andrew Howroyd_, Dec 11 2020