A390399 - OEIS

%I #17 Nov 10 2025 22:36:18

%S 1,2,6,23,109,632,4390,35621,330545,3451774,40059838,511501123,

%T 7125766429,107553198708,1748374727854,30453461219825,565835257769873,

%U 11170966770494810,233523427508804118,5153047636580080431,119697535180699705069,2919476158143732338736,74599536939513055535382

%N Number of increasing interval-labeled restricted ternary trees on the label set {0,1,...,n}.

%C An increasing interval-labeled restricted ternary tree (IRT for short) on the label set [0, n] is a vertex-labeled tree satisfying the following rules:

%C     - The underlying tree is a ternary rooted tree in which middle children do not have any siblings.

%C     - Each vertex v in the tree is assigned an interval of integer labels L_v = {i_1,...,i_2} with 0 <= i_1 <= i_2 <= n such that for each label 0 <= i <= n there exists exactly one vertex v with i contained in L_v. Thus, the collection of vertex labels forms an interval-partition of [0, n].

%C     - If a vertex v has a middle child, then |L_v| = 1; i.e., the vertex v gets a single label.

%C     - The labels are increasingly assigned; i.e., for every pair of vertices v, w such that v is the parent of w, we impose that max(L_v) < min(L_w).

%C     - The root (which is the vertex containing 0 in its label set) can only have a left child, not a middle or right child.

%C a(n) is the number of IRTs on the label set [0,n].

%H Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024.

%F G.f. (Thron-type continued fraction): 1/(1 - delta(1)*x - alpha(1)*x/(1 - delta(2)*x - alpha(2)*x/(1 - ... where delta(2k-1) = 1, delta_(2k) = k and alpha(2k-1) = alpha(2k) = k.

%e See Figure 3 in arXiv:2412.10214 for all IRTs for the label set [0,n] for n=1,2,3.

%t (*Convert a list to a T-type continued fraction.Module written by Alan Sokal.*)

%t ListToTContFrac[mylist_List, var_] := Module[{mylist2}, If[mylist == {}, Return[1]];

%t   mylist2 = mylist;

%t   If[OddQ[Length[mylist2]], AppendTo[mylist2, 0]];

%t   1/(1 - mylist2[[1]]*var - mylist2[[2]]*var*ListToTContFrac[Drop[mylist2, 2], var])]

%t myCoeffSeq[maxn_] := Flatten[Table[{1, n, n, n}, {n, 1, maxn}]]

%t mySeq[maxn_] := CoefficientList[Series[ListToTContFrac[myCoeffSeq[maxn], t], {t, 0, maxn}], t]

%K nonn

%O 0,2

%A _Bishal Deb_, Nov 04 2025