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%I #19 May 19 2021 05:14:28
%S 1,0,1,1,1,1,2,1,3,3,4,4,8,5,11,10,14,14,24,18,34,32,46,45,72,60,103,
%T 96,138,137,212,184,296,282,403,397,591,539,830,798,1125,1119,1624,
%U 1519,2253,2195,3067,3056,4341,4158,6004,5897,8145,8164,11397,11090
%N Number of series-reduced powerful rooted trees with n nodes.
%C A series-reduced rooted tree is powerful if either it is a single node, or the branches of the root all appear with multiplicities greater than 1 and are themselves series-reduced powerful rooted trees.
%H Alois P. Heinz, <a href="/A318611/b318611.txt">Table of n, a(n) for n = 1..8000</a>
%e The a(13) = 8 series-reduced powerful rooted trees:
%e ((oo)(oo)(oo)(oo))
%e ((ooo)(ooo)(ooo))
%e (ooo(oo)(oo)(oo))
%e ((ooooo)(ooooo))
%e (oo(oooo)(oooo))
%e (oooo(ooo)(ooo))
%e (oooooo(oo)(oo))
%e (oooooooooooo)
%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:= n-> `if`(n<2, n, b(n-1$2)):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Aug 31 2018
%t purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Min@@Length/@Split[#]>1&],{ptn,IntegerPartitions[n-1]}]];
%t Table[Length[purt[n]],{n,20}]
%t (* Second program: *)
%t h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, 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,
%t Sum[b[n - i*j, i - 1]*h[a[i], j, 0], {j, 0, n/i}]]];
%t a[n_] := If[n < 2, n, b[n - 1, n - 1]];
%t Array[a, 60] (* _Jean-François Alcover_, May 19 2021, after _Alois P. Heinz_ *)
%Y Cf. A000081, A001190, A001678, A001694, A004111, A167865, A291636, A317102, A317705, A317707, A318612, A318691.
%K nonn
%O 1,7
%A _Gus Wiseman_, Aug 30 2018
%E a(41)-a(56) from _Alois P. Heinz_, Aug 31 2018
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