A288942 -id:A288942 - cp's OEIS frontend

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

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 58774, 207921, 742340, 2671380, 9679341, 35283057, 129298686, 476076425, 1760356290, 6534075415, 24337242771, 90934212636, 340748853950, 1280234838924, 4821722837721, 18200855131046, 68847269742844, 260929422296290

Offset: 0

Alois P. Heinz, Sep 01 2017

nonn

Also the number of Dyck paths of semilength n with all ascent lengths <= nine.
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.
Differs from A000108 first at n = 10.

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

Column k=9 of A288942.

Cf. A000108.

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(9, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);

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, 9];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Vec(serreverse(x*(1-x)/(1-x*x^9) + O(x*x^25))) \\ Andrew Howroyd, Nov 29 2017

G.f.: G(x)/x where G(x) is the reversion of x*(1-x)/(1-x^10). - Andrew Howroyd, Dec 01 2017

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

A291825 Number of ordered rooted trees with n non-root nodes and all outdegrees <= ten.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Sep 01 2017

nonn

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 up-jumps are <= ten 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 = 11.

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

Column k=10 of A288942.

Cf. A000108.

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(10, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);

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}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

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

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

G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..10} x^k*A(x)^k. - Ilya Gutkovskiy, May 03 2019

A382060 Number of rooted ordered trees with n nodes such that the degree of each node is less than or equal to its depth plus one.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 10, 27, 77, 231, 719, 2302, 7541, 25177, 85405, 293635, 1021272, 3587674, 12713796, 45402113, 163244197, 590529759, 2147915920, 7851127319, 28826079193, 106268313333, 393218951710, 1459969448090, 5437679646441, 20311366912839, 76072367645347, 285623120079865, 1074888308119285

Offset: 0

John Tyler Rascoe, Mar 14 2025

nonn

The root vertex is depth d=0 and is to have <= d+1 = 1 children so these are "planted" trees.

			a(5) = 4 counts:
                               depth:
   o     o       o       o       0
   |     |       |       |
   o     o       o       o       1
   |     |      / \     / \
   o     o     o   o   o   o     2
   |    / \    |           |
   o   o   o   o           o     3
   |
   o                             4

John Tyler Rascoe, Python program.
Kevin Ryde, PARI/GP program

Cf. A000081, A000108, A000957, A036765, A288942, A358586, A358590, A380761.

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

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A291825 Number of ordered rooted trees with n non-root nodes and all outdegrees <= ten.

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Maple

Mathematica

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A382060 Number of rooted ordered trees with n nodes such that the degree of each node is less than or equal to its depth plus one.

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