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A318611 Number of series-reduced powerful rooted trees with n nodes. 5
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, 96, 138, 137, 212, 184, 296, 282, 403, 397, 591, 539, 830, 798, 1125, 1119, 1624, 1519, 2253, 2195, 3067, 3056, 4341, 4158, 6004, 5897, 8145, 8164, 11397, 11090 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..8000

EXAMPLE

The a(13) = 8 series-reduced powerful rooted trees:

  ((oo)(oo)(oo)(oo))

  ((ooo)(ooo)(ooo))

  (ooo(oo)(oo)(oo))

  ((ooooo)(ooooo))

  (oo(oooo)(oooo))

  (oooo(ooo)(ooo))

  (oooooo(oo)(oo))

  (oooooooooooo)

MAPLE

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

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

    end:

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

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

    end:

a:= n-> `if`(n<2, n, b(n-1$2)):

seq(a(n), n=1..60);  # Alois P. Heinz, Aug 31 2018

MATHEMATICA

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

Table[Length[purt[n]], {n, 20}]

(* Second program: *)

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}]]];

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}]]];

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

Array[a, 60] (* Jean-Fran├žois Alcover, May 19 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A000081, A001190, A001678, A001694, A004111, A167865, A291636, A317102, A317705, A317707, A318612, A318691.

Sequence in context: A112194 A238788 A083041 * A130067 A282906 A032303

Adjacent sequences:  A318608 A318609 A318610 * A318612 A318613 A318614

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 30 2018

EXTENSIONS

a(41)-a(56) from Alois P. Heinz, Aug 31 2018

STATUS

approved

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Last modified July 23 22:20 EDT 2021. Contains 346265 sequences. (Running on oeis4.)