A122774 Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows.

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

Offset: 1

Author

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

Keywords

• nonn
• tabl

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

Examples

			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
		

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.
• Index of sequences related to factorial numbers

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).

Programs

J

NB. (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)

Mathematica

Table[(2 (n - m) - 1)!! (2 (n - 1))!!/(2 (n - m))!!, {n, 8}, {m, n}] // Flatten (* Michael De Vlieger, Jan 25 2017 *)

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+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

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