A244712 Number of 2n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) n.

0, 1, 1, 3, 6, 15, 31, 74, 159, 365, 805, 1819, 4041, 9091, 20274, 45474, 101644, 227755, 509559, 1141446, 2555232, 5723626, 12817678, 28713594, 64319189, 144104857, 322867573, 723482538, 1621264326, 3633487621, 8143682973, 18253865341, 40918282628, 91730206467

Offset

0,4

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..193 (terms 0..70 from Alois P. Heinz)

Formula

a(n) = A244657(2n,n).

Maple

b:= proc(n, i, h, 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, min(i-1, h))+j-1, j)
       *b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
    end:
A:= proc(n, k) option remember;
      `if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
    end:
a:= n-> b(2*n-1$2, n$2):
seq(a(n), n=0..40);

Mathematica

b[n_, i_, h_, v_] := b[n, i, h, 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, Min[i-1, h]]+j-1, j] * b[n-i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, 1, Min[k, n-1] }]];
a[n_] := b[2n-1, 2n-1, n, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2017, after Alois P. Heinz *)

Crossrefs

Cf. A244657.

Sequence in context: A244709 A244710 A244711 * A006961 A374704 A316219

Adjacent sequences: A244709 A244710 A244711 * A244713 A244714 A244715

Keyword

nonn

Author

Alois P. Heinz, Jul 04 2014

Status

approved