A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

1, 1, 1, 2, 3, 5, 9, 16, 29, 55, 104, 200, 389, 763, 1507, 3002, 6010, 12102, 24484, 49751, 101475, 207723, 426542, 878451, 1813945, 3754918, 7790326, 16196629, 33739335, 70410401, 147187513, 308171861, 646188276, 1356847388, 2852809425, 6005542176

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Gus Wiseman, Illustration of the first 8 terms.

Example

The a(7) = 9 trees: ((((((o)))))), ((((ooo)))), (((oo(o)))), ((oo((o)))), ((o(o)(o))), (oo(((o)))), (oo(ooo)), (o(o)((o))), ((o)(o)(o)).

Maple

b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
      `if`(n=v, 1, add(binomial(a(i)+j-1, j)
       *b(n-i*j, i-1, v-j), j=0..min(n/i, v)))))
    end:
a:= n-> `if`(n<2, n, add(b(n-1$2, j), j=[1, 3])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 30 2018

Mathematica

multing[n_, k_]:=Binomial[n+k-1, k];
a[n_]:=a[n]=If[n===1, 1, Sum[Product[multing[a[x], Count[ptn, x]], {x, Union[ptn]}], {ptn, Select[IntegerPartitions[n-1], MemberQ[{1, 3}, Length[#]]&]}]];
Table[a[n], {n, 40}]
(* Second program: *)
b[n_, i_, v_] := b[n, i, v] = If[n == 0,
     If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
     If[n == v, 1, Sum[Binomial[a[i] + j - 1, j]*
     b[n - i*j, i - 1, v - j], {j, 0, Min[n/i, v]}]]]];
a[n_] := If[n < 2, n, Sum[b[n - 1, n - 1, j], {j, {1, 3}}]];
Array[a, 40] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000081, A000598, A001190, A001399, A004111, A014591, A032305, A273873, A292050, A295461, A298118, A298205.

Sequence in context: A050253 A198518 A182558 * A265581 A335703 A107250

Adjacent sequences: A298201 A298202 A298203 * A298205 A298206 A298207

Keyword

nonn

Author

Gus Wiseman, Jan 14 2018

Status

approved