|
%I #17 Jan 27 2017 13:31:57
%S 1,1,2,3,4,8,15,18,24,48,105,120,144,192,384,945,1050,1200,1440,1920,
%T 3840,10395,11340,12600,14400,17280,23040,46080,135135,145530,158760,
%U 176400,201600,241920,322560,645120
%N Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows.
%C 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).
%C 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).
%C 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.
%C ..1.. 2(n-2)+1... 7 5 3 1
%C 2(n-1).. 1 ...... 7 5 3 1
%C .........................
%C 2(n-1) 2(n-2) ... 1 5 3 1
%C 2(n-1) 2(n-2) ... 6 1 3 1
%C 2(n-1) 2(n-2) ... 6 4 1 1
%C 2(n-1) 2(n-2) ... 6 4 2 1
%H Oleg Kobchenko, <a href="http://www.jsoftware.com/jwiki/Essays/Bifactorial">Bifactorial</a>, J Wiki at jsoftware.com
%H Oleg Kobchenko, <a href="http://www.jsoftware.com/jwiki/Essays/Generalized_Monte_Hall">Generalized Monte Hall</a> problem at J Wiki
%H B. E. Meserve, <a href="http://www.jstor.org/stable/2306136">Double Factorials</a>, American Mathematical Monthly, 55 (1948), 425-426.
%H R. Ondrejka, <a href="http://dx.doi.org/10.1090/S0025-5718-70-99856-X">Tables of double factorials</a>, Math. Comp., 24 (1970), 231.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>, The World of Mathematics.
%H <a href="/index/Fa#factorial">Index of sequences related to factorial numbers</a>
%F (n B m) = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, 1<=m<=n
%F (n B 1) = (2(n-1)-1)!! = (2n-3)!!, 1<=n
%F (n B n) = (2(n-1))!!, 1<=n
%F (n B m+1) = (n B m) 2(n-m) / (2(n-m)-1), 1<=m<n
%F (n+1 B m+1) = (n B m) 2n, 1<=m<=n
%F (n+1 B m+1) = C(n,m) (2(n-m)-1)!!(2m)!!, 1<=m<=n [Corrected by _Werner Schulte_, Jan 23 2017]
%F (n+1 B 1) = Sum_{i=1..n} (n B i).
%F (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
%e Examples obtained from the expressions in J
%e 4 B 3 NB. bifactorial 4 B 3, n=4, m=3
%e 24
%e (B"0 >:(AT)i.)"0 >:i.4 NB. for 1 <= m <= n=4
%e 1 0 0 0
%e 1 2 0 0
%e 3 4 8 0
%e 15 18 24 48
%t Table[(2 (n - m) - 1)!! (2 (n - 1))!!/(2 (n - m))!!, {n, 8}, {m, n}] // Flatten (* _Michael De Vlieger_, Jan 25 2017 *)
%o In J (www.jsoftware.com):
%o Fe=: 2&^ * ! NB. even factorial, 2^n * n!
%o Fo=: !@+: % Fe NB. odd factorial, (2n)! / (2n)!!
%o B =: Fo@- * <:@[ %&Fe - NB. bifactorial, Fo(n-m) Fe(n-1) / Fe(n-m)
%Y Cf. A000165 Even factorials (2n)!! = 2^n*n!.
%Y Cf. A001147 Odd factorials (2n-1)!! = 1*3*5*...*(2n-1).
%Y Cf. A006882 Double factorials, n!!: a(n) = n*a(n-2).
%K nonn,tabl
%O 1,3
%A Oleg Kobchenko (olegyk(AT)yahoo.com), Sep 11 2006
| 