A396636 - OEIS

%I #7 Jun 07 2026 19:18:24

%S 1,3,4,21,31,32,255,420,435,436,4815,8625,8990,9011,9012,130095,

%T 250390,262045,262731,262759,262760,4763115,9755865,10241105,10269469,

%U 10270659,10270695,10270696,226955925,491081920,516730165,518213374,518275576,518277514,518277559,518277560

%N Triangle read by rows: T(n,k) is the number of at-most-k-furcating labeled histories that can be constructed for n labeled leaves, 2<=k<=n, if simultaneous branching events are allowed.

%C Labeled histories with simultaneity for at-most-k-furcating trees can be extended to n=1, with T(1,k)=1. They can also be extended to k>n, with T(n,k)=T(n,n) for k>n.

%H Emily H. Dickey and Noah A. Rosenberg, <a href="https://doi.org/10.1016/j.dam.2026.04.032">Labeled histories and maximally probable labeled topologies with multifurcation</a>, Discr. Appl. Math. 391 (2026), 192-203. See Table 2.

%e Triangle begins (the first row is n=2 and the first column is k=2):

%e         1;

%e         3;         4;

%e        21;        31;        32;

%e       255;       420;       435;       436;

%e      4815;      8625;      8990;      9011;      9012;

%e    130095;    250390;    262045;    262731;    262759;    262760;

%e   4763115;   9755865;  10241105;  10269469;  10270659;  10270695;  10270696;

%e 226955925; 491081920; 516730165; 518213374; 518275576; 518277514; 518277559; 518277560;

%Y Columns 2..6 give: A317059, A396088, A396488, A396512, A396535.

%Y Right diagonal gives A005121.

%K nonn

%O 2,2

%A _Noah A Rosenberg_, Jun 01 2026