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

%I A036769 #38 Jun 08 2019 03:56:10
%S A036769 1,1,2,5,14,42,132,429,1429,4852,16730,58422,206192,734332,2635680,
%T A036769 9524301,34622207,126520393,464517300,1712650520,6338433840,
%U A036769 23538973950,87690410580,327611738790,1227178265182,4607940112396,17341126763366,65395548619912
%N A036769 Number of ordered rooted trees with n non-root nodes and all outdegrees <= seven.
%H A036769 Alois P. Heinz, <a href="/A036769/b036769.txt">Table of n, a(n) for n = 0..1000</a>
%H A036769 Nickolas Hein and Jia Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2015.
%H A036769 Nickolas Hein and Jia Huang, <a href="https://doi.org/10.1016/j.ejc.2016.11.004">Modular Catalan Numbers</a>, European Journal of Combinatorics 61 (2017), 197-218.
%H A036769 L. Takacs, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/18_1_1.pdf">Enumeration of rooted trees and forests</a>, Math. Scientist 18 (1993), 1-10, esp. Eq. (6).
%H A036769 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A036769 G.f. A(x) satisfies: A(x) = 1 + Sum_{k=1..7} x^k*A(x)^k. - _Ilya Gutkovskiy_, May 03 2019
%p A036769 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) ];
%p A036769 # second Maple program:
%p A036769 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A036769       add(b(u-j, o+j-1), j=1..min(1, u))+
%p A036769       add(b(u+j-1, o-j), j=1..min(7, o)))
%p A036769     end:
%p A036769 a:= n-> b(0, n):
%p A036769 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 28 2017
%t A036769 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]}]];
%t A036769 a[n_] := b[0, n];
%t A036769 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 27 2017, after _Alois P. Heinz_ *)
%o A036769 (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x/sum(k=0,7,x^k)+O(x^(n+2))),n+1)) /* _Ralf Stephan_ */
%Y A036769 Column k=7 of A288942.
%K A036769 nonn
%O A036769 0,3
%A A036769 _N. J. A. Sloane_
%E A036769 Name clarified by _Andrew Howroyd_, Dec 04 2017