A182378 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)).

1, 1, 1, 2, 4, 9, 20, 48, 115, 285, 716, 1833, 4740, 12410, 32754, 87176, 233547, 629540, 1705809, 4644231, 12697500, 34848694, 95973026, 265142431, 734606478, 2040683413, 5682634446, 15859800889, 44355531103, 124290064228, 348904212741, 981082979409

Offset: 0

Author

Michael Burkhart, Apr 26 2012

Keywords

• nonn

Comments

Number of rooted trees where each node has at most 7 children.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000081, A036717, A036718, A036719, A036720, A036721, A036722, A244372, A292553, A292554, A292555, A292556.

Column k=7 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, 7$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, 7, 7]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 15 2018, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=1..7} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.338512011286603947719604869750539045616436718225097926729820... - Robert A. Russell, Feb 11 2023

Extensions

More terms from Patrick Devlin, Apr 29 2012