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A122774
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Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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Eric Weisstein's World of Mathematics, Double Factorial, The World of Mathematics.
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FORMULA
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(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
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EXAMPLE
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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
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MATHEMATICA
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Table[(2 (n - m) - 1)!! (2 (n - 1))!!/(2 (n - m))!!, {n, 8}, {m, n}] // Flatten (* Michael De Vlieger, Jan 25 2017 *)
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PROG
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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)
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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Oleg Kobchenko (olegyk(AT)yahoo.com), Sep 11 2006
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STATUS
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approved
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