A291823 Number of ordered rooted trees with n non-root nodes and all outdegrees <= eight.

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16785, 58708, 207557, 740520, 2662812, 9640581, 35112513, 128563215, 472951884, 1747233370, 6479450415, 24111470952, 90006390290, 336953657070, 1264770431964, 4758911027946, 17946417454046, 67818937355227, 256781370248500

Offset: 0

Author

Alois P. Heinz, Sep 01 2017

Keywords

• nonn

Comments

Also the number of Dyck paths of semilength n with all ascent lengths <= eight.

Also the number of permutations p of [n] such that in 0p all up-jumps are <= eight and no down-jump is larger than 1. An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

Differs from A000108 first at n = 9.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. Hein and J. Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015
• Index entries for sequences related to rooted trees

Crossrefs

Column k=8 of A288942.

Cf. A000108.

Programs

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1), j=1..min(1, u))+
      add(b(u+j-1, o-j), j=1..min(8, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);

Mathematica

b[u_, o_, k_] := b[u, o, k] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, k], {j, 1, Min[1, u]}] + Sum[b[u + j - 1, o - j, k], {j, 1, Min[k, o]}]];
a[n_] := b[0, n, 8];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

PARI

Vec(serreverse(x*(1-x)/(1-x*x^8) + O(x*x^25))) \\ Andrew Howroyd, Nov 29 2017

Formula

G.f.: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^9). - Andrew Howroyd, Nov 30 2017

G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..8} x^k*A(x)^k. - Ilya Gutkovskiy, May 03 2019