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A303259 Number of ordered rooted trees with n non-root nodes such that the maximal outdegree equals ceiling(n/2).

%I A303259 #8 Mar 19 2022 06:36:32
%S A303259 1,1,1,3,8,15,53,84,326,495,1997,3003,12370,18564,77513,116280,490306,
%T A303259 735471,3124541,4686825,20030000,30045015,129024469,193536720,
%U A303259 834451788,1251677700,5414950283,8122425444,35240152706,52860229080,229911617041,344867425584
%N A303259 Number of ordered rooted trees with n non-root nodes such that the maximal outdegree equals ceiling(n/2).
%H A303259 Alois P. Heinz, <a href="/A303259/b303259.txt">Table of n, a(n) for n = 0..2417</a>
%F A303259 a(n) = A203717(n,ceiling(n/2)).
%p A303259 b:= proc(u, o, k) option remember; `if`(u+o=0, 1,
%p A303259       add(b(u-j, o+j-1, k), j=1..min(1, u))+
%p A303259       add(b(u+j-1, o-j, k), j=1..min(k, o)))
%p A303259     end:
%p A303259 a:= n-> `if`(n=0, 1, (j-> b(0, n, j)-b(0, n, j-1))(ceil(n/2))):
%p A303259 seq(a(n), n=0..35);
%t A303259 b[u_, o_, k_] := b[u, o, k] = If[u + o == 0, 1,
%t A303259      Sum[b[u - j, o + j - 1, k], {j, 1, Min[1, u]}] +
%t A303259      Sum[b[u + j - 1, o - j, k], {j, 1, Min[k, o]}]];
%t A303259 a[n_] := If[n == 0, 1, With[{j = Ceiling[n/2]}, b[0, n, j]-b[0, n, j-1]]];
%t A303259 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Mar 19 2022, after _Alois P. Heinz_ *)
%Y A303259 Bisections give: A291662 (even part), A005809 (odd part).
%Y A303259 Cf. A203717.
%K A303259 nonn
%O A303259 0,4
%A A303259 _Alois P. Heinz_, Apr 20 2018