A396512 Number of labeled histories for rooted at-most-5-furcating trees with n leaves, if simultaneous branching events are allowed.

1, 1, 4, 32, 436, 9011, 262731, 10269469, 518213374, 32791782443, 2542620052511, 237076215199362, 26169956408743147, 3375253146527926664, 502930199914263085965, 85738904081626809938925, 16581476316696251513176624, 3610682865074032031867899316

Offset

1,3

Links

Table of n, a(n) for n=1..18.

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.

Mathematica

A[1] := 1;
A[n_] := n! Sum[(1/((2!)^x2 (x2!))) Sum[(1/((3!)^x3 (x3!))) Sum[(1/((4!)^x4 (x4!))) Sum[(1/((5!)^x5 (x5!))) (1/(n - 2 x2 - 3 x3 - 4 x4 - 5 x5)!)*A[n - x2 - 2 x3 - 3 x4 - 4 x5], {x5, Max[Ceiling[(2 - 2 x2 - 3 x3 - 4 x4)/5], 0], Floor[(n - 2 x2 - 3 x3 - 4 x4)/5]}], {x4, 0, Floor[(n - 2 x2 - 3 x3)/4]}], {x3, 0, Floor[(n - 2 x2)/3]}], {x2, 0, Floor[n/2]}];
Table[A[n], {n, 1, 15}]

Crossrefs

Cf. A317059 (2-furcating), A396088 (at-most-3-furcating), A396488 (at-most-4-furcating), A393749 (non-simultaneous branching events only).

Sequence in context: A088991 A009668 A396488 * A396535 A005121 A327379

Adjacent sequences: A396507 A396509 A396510 * A396513 A396514 A396515

Keyword

nonn

Author

Noah A Rosenberg, May 28 2026

Status

approved