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A291662 Number of ordered rooted trees with 2n non-root nodes such that the maximal outdegree equals n.

%I A291662 #21 May 15 2021 06:18:29
%S A291662 1,1,8,53,326,1997,12370,77513,490306,3124541,20030000,129024469,
%T A291662 834451788,5414950283,35240152706,229911617041,1503232609082,
%U A291662 9847379391133,64617565719052,424655979547781,2794563003870310,18412956934908669,121455445321173578
%N A291662 Number of ordered rooted trees with 2n non-root nodes such that the maximal outdegree equals n.
%C A291662 a(n) is also the number of permutations p of [2n] such that in 0p the largest up-jump equals n and no down-jump is larger than 1. An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here. a(2) = 8: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2431.
%H A291662 Alois P. Heinz, <a href="/A291662/b291662.txt">Table of n, a(n) for n = 0..1208</a>
%F A291662 a(n) = A203717(2n,n).
%F A291662 a(n) ~ (27/4)^n / sqrt(3*Pi*n). - _Vaclav Kotesovec_, May 02 2018
%e A291662 a(2) = 8:
%e A291662 .
%e A291662 .    o      o      o      o      o       o      o       o
%e A291662 .    |      |      |     / \    / \     / \    / \     / \
%e A291662 .    o      o      o    o   o  o   o   o   o  o   o   o   o
%e A291662 .    |     / \    / \   |          |  ( )        ( )  |   |
%e A291662 .    o    o   o  o   o  o          o  o o        o o  o   o
%e A291662 .   / \   |          |  |          |
%e A291662 .  o   o  o          o  o          o
%t A291662 b[n_, t_, k_] := b[n, t, k] = If[n == 0, 1, If[t > 0, Sum[b[j - 1, k, k]* b[n - j, t - 1, k], {j, 1, n}], b[n - 1, k, k]]];
%t A291662 T[n_, k_] := b[n, k - 1, k - 1] - If[k == 1, 0, b[n, k - 2, k - 2]];
%t A291662 a[n_] := T[2n, n];
%t A291662 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 29 2019, after _Alois P. Heinz_ in A203717 *)
%o A291662 (Python)
%o A291662 from sympy.core.cache import cacheit
%o A291662 @cacheit
%o A291662 def b(u, o, k): return 1 if u + o==0 else sum([b(u - j, o + j - 1, k) for j in range(1, min(1, u) + 1)]) + sum([b(u + j - 1, o - j, k) for j in range(1, min(k, o) + 1)])
%o A291662 def a(n): return b(0, 2*n, n) - (0 if n==0 else b(0, 2*n, n - 1))
%o A291662 print([a(n) for n in range(31)]) # _Indranil Ghosh_, Aug 30 2017
%Y A291662 Cf. A203717.
%Y A291662 Bisection (even part) of A303259.
%K A291662 nonn
%O A291662 0,3
%A A291662 _Alois P. Heinz_, Aug 28 2017