A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)).

1, 1, 1, 2, 4, 9, 20, 47, 112, 277, 693, 1766, 4547, 11852, 31146, 82534, 220149, 590834, 1593951, 4320723, 11761394, 32138301, 88121176, 242383729, 668607115, 1849194691, 5126800907, 14245679652, 39666239726, 110661514973, 309280533011, 865839831118

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also the number of rooted trees where each node has at most 5 children. [Patrick Devlin, Apr 30 2012]

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Joerg Arndt, The a(6) = 20 rooted trees with 5 nodes and out-degrees <= 5

Crossrefs

Cf. A000081, A036717, A036718, A036719, A036720, A036722, A182378, A244372, A292553, A292554, A292555, A292556.

Column k=5 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, 5$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, 5, 5]];
Table[a[n], {n, 0, 35}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=1..5} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.340017469151060086823930137816585262710976835711484267209811... - Robert A. Russell, Feb 11 2023