cp's OEIS Frontend

A291824 Number of ordered rooted trees with n non-root nodes and all outdegrees <= nine.

%I A291824 #26 May 03 2019 21:23:35
%S A291824 1,1,2,5,14,42,132,429,1430,4862,16795,58774,207921,742340,2671380,
%T A291824 9679341,35283057,129298686,476076425,1760356290,6534075415,
%U A291824 24337242771,90934212636,340748853950,1280234838924,4821722837721,18200855131046,68847269742844,260929422296290
%N A291824 Number of ordered rooted trees with n non-root nodes and all outdegrees <= nine.
%C A291824 Also the number of Dyck paths of semilength n with all ascent lengths <= nine.
%C A291824 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 A291824 Differs from A000108 first at n = 10.
%H A291824 Alois P. Heinz, <a href="/A291824/b291824.txt">Table of n, a(n) for n = 0..1000</a>
%H A291824 N. Hein and J. Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015
%H A291824 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A291824 G.f.: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^10). - _Andrew Howroyd_, Dec 01 2017
%F A291824 G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..9} x^k*A(x)^k. - _Ilya Gutkovskiy_, May 03 2019
%p A291824 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A291824       add(b(u-j, o+j-1), j=1..min(1, u))+
%p A291824       add(b(u+j-1, o-j), j=1..min(9, o)))
%p A291824     end:
%p A291824 a:= n-> b(0, n):
%p A291824 seq(a(n), n=0..30);
%t A291824 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 A291824 a[n_] := b[0, n, 9];
%t A291824 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)
%o A291824 (PARI) Vec(serreverse(x*(1-x)/(1-x*x^9) + O(x*x^25))) \\ _Andrew Howroyd_, Nov 29 2017
%Y A291824 Column k=9 of A288942.
%Y A291824 Cf. A000108.
%K A291824 nonn
%O A291824 0,3
%A A291824 _Alois P. Heinz_, Sep 01 2017