A291824 - OEIS

%I #26 May 03 2019 21:23:35

%S 1,1,2,5,14,42,132,429,1430,4862,16795,58774,207921,742340,2671380,

%T 9679341,35283057,129298686,476076425,1760356290,6534075415,

%U 24337242771,90934212636,340748853950,1280234838924,4821722837721,18200855131046,68847269742844,260929422296290

%N Number of ordered rooted trees with n non-root nodes and all outdegrees <= nine.

%C Also the number of Dyck paths of semilength n with all ascent lengths <= nine.

%C Also the number of permutations p of [n] such that in 0p all up-jumps are <= nine 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 Differs from A000108 first at n = 10.

%H Alois P. Heinz, <a href="/A291824/b291824.txt">Table of n, a(n) for n = 0..1000</a>

%H N. Hein and J. Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F G.f.: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^10). - _Andrew Howroyd_, Dec 01 2017

%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..9} x^k*A(x)^k. - _Ilya Gutkovskiy_, May 03 2019

%p b:= proc(u, o) option remember; `if`(u+o=0, 1,

%p       add(b(u-j, o+j-1), j=1..min(1, u))+

%p       add(b(u+j-1, o-j), j=1..min(9, o)))

%p     end:

%p a:= n-> b(0, n):

%p seq(a(n), n=0..30);

%t 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 a[n_] := b[0, n, 9];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)

%o (PARI) Vec(serreverse(x*(1-x)/(1-x*x^9) + O(x*x^25))) \\ _Andrew Howroyd_, Nov 29 2017

%Y Column k=9 of A288942.

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 01 2017