

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


REFERENCES

B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425426.
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(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(m,n)(2(nm)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(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 * A189740 A126294 A224912
Adjacent sequences: A122771 A122772 A122773 * A122775 A122776 A122777


KEYWORD

nonn,tabl


AUTHOR

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


STATUS

approved



