OFFSET
1,3
COMMENTS
Bifactorial (n B m) is the number of ways of drawing the single marked item in choice m out of n choices with n-1 alternating draws of unmarked items, both without replacement, out of 2n-1 total items. Probability P(m|n) of drawing the marked item in choice m of n total choices is P(m|n) = (n B m) / (n+1 B 1).
Generalized Monte Hall (GMH) problem: There are 2n-1 doors, behind each door there is either a car or one of 2n-2 goats. Player picks a door (Choice 1), game master reveals another door with a goat. Player can either stay with Choice 1 or continue to play. In which case he chooses one of the 2n-3 remaining doors (Choice 2). Game master then reveals another door with a goat and the player can either stay with Choice 2 or continue to play the same way till the last door (Choice n). Number of ways to pick a car in Choice m out of n total choices is (n B m).
The name "bifactorial" comes from triangular matrix of rank n, with even factorials in the lower half and odd ones in the upper, whose products by m-th rows gives n B m. Such matrix describes the sample space of outcomes in GMH for each choice i given car in choice m.
..1.. 2(n-2)+1... 7 5 3 1
2(n-1).. 1 ...... 7 5 3 1
.........................
2(n-1) 2(n-2) ... 1 5 3 1
2(n-1) 2(n-2) ... 6 1 3 1
2(n-1) 2(n-2) ... 6 4 1 1
2(n-1) 2(n-2) ... 6 4 2 1
LINKS
Oleg Kobchenko, Bifactorial, J Wiki at jsoftware.com
Oleg Kobchenko, Generalized Monte Hall problem at J Wiki
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
Eric Weisstein's World of Mathematics, Double Factorial, The World of Mathematics.
FORMULA
(n B m) = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, 1<=m<=n
(n B 1) = (2(n-1)-1)!! = (2n-3)!!, 1<=n
(n B n) = (2(n-1))!!, 1<=n
(n B m+1) = (n B m) 2(n-m) / (2(n-m)-1), 1<=m<n
(n+1 B m+1) = (n B m) 2n, 1<=m<=n
(n+1 B m+1) = C(n,m) (2(n-m)-1)!!(2m)!!, 1<=m<=n [Corrected by Werner Schulte, Jan 23 2017]
(n+1 B 1) = Sum_{i=1..n} (n B i).
(n B m) = binomial(2*n-2*m,n-m)*((n-1)!)/2^(n+1-2*m) for 1<=m<=n. - Werner Schulte, Jan 23 2017
EXAMPLE
Examples obtained from the expressions in J
4 B 3 NB. bifactorial 4 B 3, n=4, m=3
24
(B"0 >:(AT)i.)"0 >:i.4 NB. for 1 <= m <= n=4
1 0 0 0
1 2 0 0
3 4 8 0
15 18 24 48
MATHEMATICA
Table[(2 (n - m) - 1)!! (2 (n - 1))!!/(2 (n - m))!!, {n, 8}, {m, n}] // Flatten (* Michael De Vlieger, Jan 25 2017 *)
PROG
In J (www.jsoftware.com):
Fe=: 2&^ * ! NB. even factorial, 2^n * n!
Fo=: !@+: % Fe NB. odd factorial, (2n)! / (2n)!!
B =: Fo@- * <:@[ %&Fe - NB. bifactorial, Fo(n-m) Fe(n-1) / Fe(n-m)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Oleg Kobchenko (olegyk(AT)yahoo.com), Sep 11 2006
STATUS
approved