A381486 -id:A381486 - cp's OEIS frontend

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

Noah A Rosenberg, Feb 26 2025

nonn

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

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

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

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.

A381533 Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are allowed.

Original entry on oeis.org

1, 1, 126, 198198, 1552358808, 41269930621920, 2917021792126858416, 466738566750935966462976, 150642168106131265276308435840, 89930728809765858827345682838905216, 92814015425659158860323886440105229380608, 156870775305420194841270876582071899442900414976, 415352074564676036635314305973768435826840253066044416

Offset: 0

Noah A Rosenberg, Feb 26 2025

nonn

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

Cf. A317059, A381486, A381523.

a:= proc(n) option remember; `if`(n=0, 1, add((4*n+1)!/
      (i!*120^i*(4*n+1-5*i)!)*a(n-i), i=1..(4*n+1)/5))
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 26 2025

a(n) = Y(4n+1), where Y(n) = Sum_{i=1..floor(n/5)} (n!/(i!*120^i*(n-5*i)!)) * Y(n-4*i), with Y(1)=1.

cp's OEIS Frontend

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

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