

A291825


Number of ordered rooted trees with n nonroot nodes and all outdegrees <= ten.


2



1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58785, 207999, 742795, 2673760, 9690969, 35337321, 129543843, 477158000, 1765043115, 6554105415, 24421914855, 91289026931, 342225162126, 1286341683924, 4846861938006, 18303921153521, 69268371485362, 262644901975126
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OFFSET

0,3


COMMENTS

Also the number of Dyck paths of semilength n with all ascent lengths <= ten.
Also the number of permutations p of [n] such that in 0p all upjumps are <= ten and no downjump is larger than 1. An upjump j occurs at position i in p if p_{i} > p_{i1} 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_{i1}. A downjump j occurs at position i in p if p_{i} < p_{i1} 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_{i1}. First index in the lists is 1 here.
Differs from A000108 first at n = 11.


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


FORMULA

G.f.: G(x)/x where G(x) is the reversion of x*(1x)/(1x^11).  Andrew Howroyd, Dec 01 2017


MAPLE

b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(uj, o+j1), j=1..min(1, u))+
add(b(u+j1, oj), j=1..min(10, 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, 10];
Table[a[n], {n, 0, 30}] (* JeanFrançois Alcover, Nov 07 2017, after Alois P. Heinz *)


PROG

(PARI) Vec(serreverse(x*(1x)/(1x*x^10) + O(x*x^25))) \\ Andrew Howroyd, Nov 29 2017


CROSSREFS

Column k=10 of A288942.
Cf. A000108.
Sequence in context: A261592 A291824 A287973 * A287974 A115140 A120588
Adjacent sequences: A291822 A291823 A291824 * A291826 A291827 A291828


KEYWORD

nonn


AUTHOR

Alois P. Heinz, Sep 01 2017


STATUS

approved



