cp's OEIS Frontend

A180967 Number of n-game win/loss series that contain at least one dead game.

%I A180967 #30 Sep 20 2014 02:36:42
%S A180967 0,0,4,4,20,24,88,116,372,520,1544,2248,6344,9520,25904,39796,105332,
%T A180967 164904,427048,679064,1727640,2783440,6977744,11368904,28146120,
%U A180967 46307664,113416528,188202256,456637712,763506784
%N A180967 Number of n-game win/loss series that contain at least one dead game.
%C A180967 A series of n games are played between two teams. The outcome of each game is either a win or a loss (there are no draws). A team wins the whole series if it wins k=floor(n/2)+1 games or more. If a team reaches k wins then the games that follow (if there are any) are dead games, because their outcome cannot affect the outcome of the series.
%C A180967 Number of n-game series whose outcome is decided in the last game is A063886(n).
%F A180967 The last game is "alive" if and only if the result of the first n-1 games
%F A180967 is either (if n is odd) (n-1)/2 wins for both teams, or (if n is even) n/2 wins for one and n/2-1 for the other. Hence a(n)=2^n - 2C(n-1,(n-1)/2) for odd n and a(n)=2^n - 4C(n-1,n/2) for even n. - _Robert Israel_, Jan 28 2011
%F A180967 -n*a(n) +n*a(n-1) +2*(3*n-5)*a(n-2) +4*(-n+1)*a(n-3) +8*(-n+4)*a(n-4)=0. - _R. J. Mathar_, May 19 2014
%e A180967 We can represent an n-game series as a binary string of length n, where '0' means a loss for the first team and '1' means a win for the first team. For n=3 there are 2^3=8 possible game series. Out of these there are 4 that contain at least one dead game (the last one): 000, 001, 110, 111. Hence a(3)=4.
%t A180967 f[n_] := 2^n - 2*If[ OddQ@ n, Binomial[n - 1, (n - 1)/2], 2 Binomial[n - 1, n/2]]; Array[f, 30] (* _Robert G. Wilson v_ *)
%Y A180967 See A181618 for win/loss/draw series.
%K A180967 nonn
%O A180967 1,3
%A A180967 _Dmitry Kamenetsky_, Jan 28 2011