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
Keywords
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
Crossrefs
Column k=6 of A288942.
Programs
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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 *) -
PARI
a(n)=if(n<0,0,polcoeff(serreverse(x/polcyclo(7)+O(x^(n+2))),n+1)) /* Ralf Stephan */
Formula
G.f. A(x) satisfies A(x)=1+sum(n=1..6, (x*A(x))^n). - Vladimir Kruchinin, Feb 22 2011
Extensions
Name clarified by Andrew Howroyd, Dec 04 2017