A381523 Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are allowed.

1, 1, 35, 8925, 8033025, 19010866875, 97622651251125, 958647115051250625, 16437666902498106890625, 459581350409578975249546875, 19861812620603175030206132109375, 1271123241419341933758758697996796875, 116303414318027015186301064741488195703125, 14773177703549629967524262172307456486365234375

Offset: 0

Author

Noah A Rosenberg, Feb 26 2025

Keywords

• nonn

Links

• Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Table 4)

Crossrefs

Cf. A317059 for binary trees and A381486 for ternary trees.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add((3*n+1)!/
      (i!*24^i*(3*n+1-4*i)!)*a(n-i), i=1..(3*n+1)/4))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Feb 26 2025

Formula

a(n) = Y(3n+1), where Y(n) = Sum_{i=1..floor(n/4)} (n!/(i!*24^i*(n-4*i)!))*Y(n-3*i), with Y(1)=1.