A396636 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.

1, 3, 4, 21, 31, 32, 255, 420, 435, 436, 4815, 8625, 8990, 9011, 9012, 130095, 250390, 262045, 262731, 262759, 262760, 4763115, 9755865, 10241105, 10269469, 10270659, 10270695, 10270696, 226955925, 491081920, 516730165, 518213374, 518275576, 518277514, 518277559, 518277560

Offset

2,2

Comments

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.

Links

Table of n, a(n) for n=2..37.

Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, Discr. Appl. Math. 391 (2026), 192-203. See Table 2.

Example

Triangle begins (the first row is n=2 and the first column is k=2):
        1;
        3;         4;
       21;        31;        32;
      255;       420;       435;       436;
     4815;      8625;      8990;      9011;      9012;
   130095;    250390;    262045;    262731;    262759;    262760;
  4763115;   9755865;  10241105;  10269469;  10270659;  10270695;  10270696;
226955925; 491081920; 516730165; 518213374; 518275576; 518277514; 518277559; 518277560;

Crossrefs

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

Right diagonal gives A005121.

Sequence in context: A339482 A359883 A032830 * A225478 A254884 A034475

Adjacent sequences: A396633 A396634 A396635 * A396637 A396638 A396639

Keyword

nonn

Author

Noah A Rosenberg, Jun 01 2026

Status

approved