

A122774


Triangle of bifactorial numbers, n B m = (2(nm)1)!! (2(n1))!! / (2(nm))!!, 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
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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 n1 alternating draws of unmarked items, both without replacement, out of 2n1 total items. Probability P(mn) of drawing the marked item in choice m of n total choices is P(mn) = (n B m) / (n+1 B 1).
Generalized Monte Hall (GMH) problem: There are 2n1 doors, behind each door there is either a car or one of 2n2 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 2n3 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 mth 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(n2)+1... 7 5 3 1
2(n1).. 1 ...... 7 5 3 1
.........................
2(n1) 2(n2) ... 1 5 3 1
2(n1) 2(n2) ... 6 1 3 1
2(n1) 2(n2) ... 6 4 1 1
2(n1) 2(n2) ... 6 4 2 1


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
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425426.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
Eric Weisstein's World of Mathematics, Double Factorial, The World of Mathematics.
Index of sequences related to factorial numbers


FORMULA

(n B m) = (2(nm)1)!! (2(n1))!! / (2(nm))!!, 1<=m<=n
(n B 1) = (2(n1)1)!! = (2n3)!!, 1<=n
(n B n) = (2(n1))!!, 1<=n
(n B m+1) = (n B m) 2(nm) / (2(nm)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(nm)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*n2*m,nm)*((n1)!)/2^(n+12*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(nm) Fe(n1) / Fe(nm)


CROSSREFS

Cf. A000165 Even factorials (2n)!! = 2^n*n!.
Cf. A001147 Odd factorials (2n1)!! = 1*3*5*...*(2n1).
Cf. A006882 Double factorials, n!!: a(n) = n*a(n2).
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



