A292553 Number of rooted unlabeled trees on n nodes where each node has at most 8 children.

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1839, 4757, 12460, 32897, 87592, 234746, 633013, 1715851, 4673320, 12781759, 35093010, 96681705, 267199518, 740580555, 2058042803, 5733101603, 16006590851, 44782679547, 125533577578, 352525803976, 991634575368

Offset: 0

Author

Marko Riedel, Sep 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Marko Riedel, Trees with bounded degree.
• Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 1).
• Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 2)
• Marko Riedel, Maple code (FEQ 2) optimized for speed.

Crossrefs

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292554, A292555, A292556.

Column k=8 of A299038.

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(n-1$2, 8$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

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]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 8, 8]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

Functional equation of G.f. is T(z) = z + z*Sum_{q=1..8} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is T(z) = 1 + z*Z(S_8)(T(z)).

a(n) = Sum_{j=1..8} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017

a(n) / a(n+1) ~ 0.338386042364849957035744926227166370702775721795018600630554... - Robert A. Russell, Feb 11 2023