%I #21 Mar 18 2026 03:48:56

%S 1,1,2,3,5,6,11,13,22,29,46,57,94,115,180,230,349,435,671,830,1245,

%T 1572,2320,2894,4287,5328,7773,9752,14066,17547,25328,31515,45010,

%U 56289,79805,99467,140778,175215,246278,307273,429421,534774,745776,927776,1287038

%N Number of powerful rooted trees with n nodes.

%C An unlabeled rooted tree is powerful if either it is a single node or a single node with a single powerful tree as a branch, or if the branches of the root all appear with multiplicities greater than 1 and are themselves powerful trees.

%H Alois P. Heinz, <a href="/A317707/b317707.txt">Table of n, a(n) for n = 1..8000</a>

%e The a(7) = 11 powerful rooted trees:

%e ((((((o))))))

%e (((((oo)))))

%e ((((ooo))))

%e ((((o)(o))))

%e (((oooo)))

%e ((ooooo))

%e (((o))((o)))

%e ((oo)(oo))

%e ((o)(o)(o))

%e (oo(o)(o))

%e (oooooo)

%p h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),

%p `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))

%p end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))

%p end:

%p a:= proc(n) option remember; `if`(n<2, n, b(n-1$2)+a(n-1)) end:

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Aug 31 2018

%t purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,Min@@Length/@Split[#]>1]&],{ptn,IntegerPartitions[n-1]}]];

%t Table[Length[purt[n]],{n,10}]

%t (* Alternative: *)

%t h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];

%t a[n_] := a[n] = If[n < 2, n, b[n - 1, n - 1] + a[n - 1]];

%t Array[a, 50] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)

%Y Cf. A000081, A001190, A001694, A004111, A301700, A303431, A317102.

%Y Cf. A317705, A317708, A317709, A317710, A317711, A317712, A317718, A317719.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 05 2018

%E a(27)-a(45) from _Alois P. Heinz_, Aug 31 2018