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

%I A291823 #30 May 03 2019 21:23:29
%S A291823 1,1,2,5,14,42,132,429,1430,4861,16785,58708,207557,740520,2662812,
%T A291823 9640581,35112513,128563215,472951884,1747233370,6479450415,
%U A291823 24111470952,90006390290,336953657070,1264770431964,4758911027946,17946417454046,67818937355227,256781370248500
%N A291823 Number of ordered rooted trees with n non-root nodes and all outdegrees <= eight.
%C A291823 Also the number of Dyck paths of semilength n with all ascent lengths <= eight.
%C A291823 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.
%C A291823 Differs from A000108 first at n = 9.
%H A291823 Alois P. Heinz, <a href="/A291823/b291823.txt">Table of n, a(n) for n = 0..1000</a>
%H A291823 N. Hein and J. Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015
%H A291823 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A291823 G.f.: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^9). - _Andrew Howroyd_, Nov 30 2017
%F A291823 G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..8} x^k*A(x)^k. - _Ilya Gutkovskiy_, May 03 2019
%p A291823 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A291823       add(b(u-j, o+j-1), j=1..min(1, u))+
%p A291823       add(b(u+j-1, o-j), j=1..min(8, o)))
%p A291823     end:
%p A291823 a:= n-> b(0, n):
%p A291823 seq(a(n), n=0..30);
%t A291823 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]}]];
%t A291823 a[n_] := b[0, n, 8];
%t A291823 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)
%o A291823 (PARI) Vec(serreverse(x*(1-x)/(1-x*x^8) + O(x*x^25))) \\ _Andrew Howroyd_, Nov 29 2017
%Y A291823 Column k=8 of A288942.
%Y A291823 Cf. A000108.
%K A291823 nonn
%O A291823 0,3
%A A291823 _Alois P. Heinz_, Sep 01 2017