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A122774 Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows. 1
1, 1, 2, 3, 4, 8, 15, 18, 24, 48, 105, 120, 144, 192, 384, 945, 1050, 1200, 1440, 1920, 3840, 10395, 11340, 12600, 14400, 17280, 23040, 46080, 135135, 145530, 158760, 176400, 201600, 241920, 322560, 645120 (list; table; graph; refs; listen; history; text; internal format)
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

REFERENCES

B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.

R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.

LINKS

Table of n, a(n) for n=1..36.

Oleg Kobchenko, Bifactorial, J Wiki at jsoftware.com

Oleg Kobchenko, Generalized Monte Hall problem at J Wiki

Eric Weisstein's World of Mathematics, Double Factorial, The World of Mathematics.

Index of sequences related to factorial numbers

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(m,n)(2(n-m)-1)!!(2m)!! 2n, 1<=m<=n, binomial relation

(n+1 B 1) = sum_{i=1..n} (n B i)

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

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

Cf. A000165 Even factorials (2n)!! = 2^n*n!.

Cf. A001147 Odd factorials (2n-1)!! = 1.3.5..(2n-1).

Cf. A006882 Double factorials, n!!: a(n)=n*a(n-2).

Sequence in context: A005853 A161460 A097029 * A274166 A189740 A126294

Adjacent sequences:  A122771 A122772 A122773 * A122775 A122776 A122777

KEYWORD

nonn,tabl

AUTHOR

Oleg Kobchenko (olegyk(AT)yahoo.com), Sep 11 2006

STATUS

approved

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Last modified December 8 12:55 EST 2016. Contains 278945 sequences.