A036722 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(6), A(x)).

1, 1, 1, 2, 4, 9, 20, 48, 114, 283, 710, 1816, 4690, 12267, 32338, 85978, 230080, 619521, 1676808, 4560286, 12454272, 34143682, 93928091, 259208006, 717375068, 1990625390, 5537142610, 15436744525, 43124847431, 120708508008, 338477040445, 950714584576

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is also the number of rooted trees where each node has at most 6 children. [Patrick Devlin, Apr 29 2012]

Links

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

Crossrefs

Cf. A000081, A036717, A036718, A036719, A036720, A036721, A182378, A244372, A292553, A292554, A292555, A292556.

Column k=6 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, 6$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, 6, 6]];
Table[a[n] , {n, 0, 35}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=1..6} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.338887196052856714304749078960983936661485522864792573284374... - Robert A. Russell, Feb 11 2023