A167484 - OEIS

%I #35 Aug 29 2021 18:54:12

%S 1,1,6,108,4320,324000,40824000,8001504000,2304433152000,

%T 933295426560000,513312484608000000,372664863825408000000,

%U 348814312540581888000000,412647331735508373504000000,606591577651197309050880000000,1091864839772155156291584000000000,2375897891344209620090486784000000000

%N For n people on one side of a river, the number of ways they can all travel to the opposite side following the pattern of 2 sent, 1 returns, 2 sent, 1 returns, ..., 2 sent.

%C This problem might arise if there was only a two-person boat available.

%C Also the number of ranked tree-child networks. - _Michael Fuchs_, May 29 2021

%H Michael De Vlieger, <a href="/A167484/b167484.txt">Table of n, a(n) for n = 1..190</a>

%H Francois Bienvenu, Amaury Lambert, and Mike Steel, <a href="https://arxiv.org/abs/2007.09701">Combinatorial and stochastic properties of ranked tree-child networks</a>, arXiv:2007.09701 [math.PR], 2021.

%H Alessandra Caraceni, Michael Fuchs, and Guan-Ru Yu, <a href="https://arxiv.org/abs/2105.10137">Bijections for ranked tree-child networks</a>, arXiv:2105.10137 [math.CO], 2021.

%F a(n) = n!*((n-1)!)^2/((2!)^(n-1)).

%F a(n) ~ 4*sqrt(2)*Pi^(3/2)*n^(3*n-1/2)/(2^n*exp(3*n)). - _Ilya Gutkovskiy_, Dec 17 2016

%e For n=3 there are 6 ways. Let a,b,c start on one side. We have:

%e 1) Send (a,b), return(a), send(a,c);

%e 2) Send (a,b), return(b), send(b,c);

%e 3) Send (b,c), return(b), send(a,b);

%e 4) Send (b,c), return(c), send(a,c);

%e 5) Send (a,c), return(a), send(a,b);

%e 6) Send (a,c), return(c), send(b,c).

%t f[n_] := n! (n - 1)!^2/2^(n - 1); Array[f, 15] (* _Robert G. Wilson v_, Dec 17 2016 *)

%K easy,nonn

%O 1,3

%A Ron Smith (ron.smith(AT)henryschein.com), Nov 04 2009

%E a(13) and a(14) corrected by _Ilya Gutkovskiy_, Dec 17 2016

%E More terms from _Ilya Gutkovskiy_, Dec 18 2016