A244410 Number of unlabeled rooted trees with 2n+1 nodes and maximal outdegree (branching factor) n.

1, 1, 5, 16, 49, 142, 415, 1198, 3473, 10048, 29118, 84376, 244747, 710198, 2062273, 5991417, 17416400, 50652247, 147384675, 429043389, 1249508946, 3640449678, 10610613551, 30937605075, 90237313082, 263288153073, 768449666116, 2243530461066, 6552016136666

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Jun 27 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A244372, A244407, A051491.

Programs

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n$2, n$2)-b(2*n$2, n-1$2)):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := If[n == 0, 1, b[2*n, 2 n, n, n] - b[2*n, 2 n, n - 1, n - 1]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) = A244372(2n+1,n).

a(n) ~ c * d^n / sqrt(n), where d = 2.955765285651994974714817524... is the Otter's rooted tree constant (see A051491), and c = 2.806733... . - Vaclav Kotesovec, Jul 11 2014