A036768 Number of ordered rooted trees with n non-root nodes and all outdegrees <= six.

1, 1, 2, 5, 14, 42, 132, 428, 1421, 4807, 16510, 57421, 201824, 715768, 2558167, 9204651, 33315919, 121218195, 443107245, 1626546453, 5993256280, 22158739970, 82182744284, 305670888560, 1139892935454, 4261095044346, 15964169665031, 59933390160322

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

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.

Lajos Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (6).

Index entries for sequences related to rooted trees

Formula

G.f. A(x) satisfies A(x)=1+sum(n=1..6, (x*A(x))^n). - Vladimir Kruchinin, Feb 22 2011

Maple

r := 6; [ 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(6, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 28 2017

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, 6];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Prog

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

Crossrefs

Column k=6 of A288942.

Sequence in context: A152226 A054393 A261589 * A287970 A058094 A080938

Adjacent sequences: A036765 A036766 A036767 * A036769 A036770 A036771

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Name clarified by Andrew Howroyd, Dec 04 2017

Status

approved