

A180967


Number of ngame win/loss series that contain at least one dead game.


1



0, 0, 4, 4, 20, 24, 88, 116, 372, 520, 1544, 2248, 6344, 9520, 25904, 39796, 105332, 164904, 427048, 679064, 1727640, 2783440, 6977744, 11368904, 28146120, 46307664, 113416528, 188202256, 456637712, 763506784
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OFFSET

1,3


COMMENTS

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.
Number of ngame series whose outcome is decided in the last game is A063886(n).


LINKS

Table of n, a(n) for n=1..30.


FORMULA

The last game is "alive" if and only if the result of the first n1 games
is either (if n is odd) (n1)/2 wins for both teams, or (if n is even) n/2 wins for one and n/21 for the other. Hence a(n)=2^n  2C(n1,(n1)/2) for odd n and a(n)=2^n  4C(n1,n/2) for even n.  Robert Israel, Jan 28 2011
n*a(n) +n*a(n1) +2*(3*n5)*a(n2) +4*(n+1)*a(n3) +8*(n+4)*a(n4)=0.  R. J. Mathar, May 19 2014


EXAMPLE

We can represent an ngame 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.


MATHEMATICA

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 *)


CROSSREFS

See A181618 for win/loss/draw series.
Sequence in context: A087213 A117857 A165559 * A231884 A052923 A014433
Adjacent sequences: A180964 A180965 A180966 * A180968 A180969 A180970


KEYWORD

nonn


AUTHOR

Dmitry Kamenetsky, Jan 28 2011


STATUS

approved



