A036769 Number of ordered rooted trees with n non-root nodes and all outdegrees <= seven.

1, 1, 2, 5, 14, 42, 132, 429, 1429, 4852, 16730, 58422, 206192, 734332, 2635680, 9524301, 34622207, 126520393, 464517300, 1712650520, 6338433840, 23538973950, 87690410580, 327611738790, 1227178265182, 4607940112396, 17341126763366, 65395548619912

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Nickolas Hein and Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015.
• Nickolas Hein and Jia Huang, Modular Catalan Numbers, European Journal of Combinatorics 61 (2017), 197-218.
• L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (6).
• Index entries for sequences related to rooted trees

Crossrefs

Column k=7 of A288942.

Programs

Maple

r := 7; [ seq((1/n)*add( (-1)^j*binomial(n,j)*binomial(2*n-2-j*(r+1), n-1),j=0..floor((n-1)/(r+1))), n=1..30) ];
# second Maple program:
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(7, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 28 2017

Mathematica

b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, Min[1, u]}] + Sum[b[u + j - 1, o - j], {j, 1, Min[7, o]}]];
a[n_] := b[0, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 27 2017, after Alois P. Heinz *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse(x/sum(k=0,7,x^k)+O(x^(n+2))),n+1)) /* Ralf Stephan */

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..7} x^k*A(x)^k. - Ilya Gutkovskiy, May 03 2019

Extensions

Name clarified by Andrew Howroyd, Dec 04 2017