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A320453 a(n) = (n^n + n*(-1)^n)/(n + 1).

%I A320453 #36 Sep 08 2022 08:46:23
%S A320453 0,2,6,52,520,6666,102942,1864136,38742048,909090910,23775972550,
%T A320453 685853880636,21633936185160,740800455037202,27368368148803710,
%U A320453 1085102592571150096,45957792327018709120,2070863582910344082918,98920982783015679456198,4993219047619047619047620
%N A320453 a(n) = (n^n + n*(-1)^n)/(n + 1).
%C A320453 In a game in which n+1 players are passing a ball from one to another, a(n) is the number of ways that the ball can start at a given player and, on the n-th pass, return (not necessarily for the first time) to that same player. E.g., the a(3)=6 ways are A-BCA, A-CBA, A-BDA, A-DBA, A-CDA, A-DCA.
%H A320453 Muniru A Asiru, <a href="/A320453/b320453.txt">Table of n, a(n) for n = 1..385</a>
%F A320453 a(n) = (n^n + n*(-1)^n)/(n + 1).
%F A320453 a(n) = A000312(n+1) - A185634(n) = A185634(n)/n + (-1)^n.
%p A320453 a:=n->(n^n+n*(-1)^n)/(n+1): seq(a(n),n=1..20); # _Muniru A Asiru_, Oct 13 2018
%t A320453 Table[(n^n + n*(-1)^n)/(n + 1), {n, 0, 50}]
%o A320453 (PARI) a(n) = (n^n + n*(-1)^n)/(n + 1);
%o A320453 (GAP) List([1..20],n->(n^n+n*(-1)^n)/(n+1)); # _Muniru A Asiru_, Oct 13 2018
%o A320453 (Magma) [(n^n + n*(-1)^n)/(n + 1) : n in [1..25]]; // _Vincenzo Librandi_, Oct 14 2018
%Y A320453 Cf. A185634, A000312.
%K A320453 nonn
%O A320453 1,2
%A A320453 _Jinyuan Wang_, Oct 13 2018
%E A320453 a(10)-a(20) from _Muniru A Asiru_, Oct 13 2018