A164705 T(n,k)=Binomial(2n-k,n)*2^(k-1) for n>=1,0<=k<=n

1, 1, 3, 3, 2, 10, 10, 8, 4, 35, 35, 30, 20, 8, 126, 126, 112, 84, 48, 16, 462, 462, 420, 336, 224, 112, 32, 1716, 1716, 1584, 1320, 960, 576, 256, 64, 6435, 6435, 6006, 5148, 3960, 2640, 1440, 576, 128, 24310, 24310, 22880, 20020, 16016, 11440, 7040, 3520, 1280

Offset

1,3

Comments

T(n,k) is the number of 2n digit binary sequences in which the (n+1)th zero occurs in the (2n-k+1)th position. T(n,k)/2^(2n-1) is the probability sought in Banach's matchbox problem. Row sum is 2^(2n-1) T(n,0)=T(n,1)=A001700(n)

Links

Table of n, a(n) for n=1..53.

Example

T(2,1)=3 because there are 3 length 4 binary sequences in which the third zero appears in the fourth position: {0,0,1,0},{0,1,0,0},{1,0,0,0}.
Triangle begins
 1,   1
 3,   3,   2
 10,  10,  8,   4
 35,  35,  30,  20, 8
 126, 126, 112, 84, 48, 16

Mathematica

Table[Table[Binomial[2 n - k, n]*2^(k - 1), {k, 0, n}], {n, 1, 9}] // Grid

Crossrefs

Sequence in context: A250304 A330307 A256916 * A073754 A193229 A112458

Adjacent sequences: A164702 A164703 A164704 * A164706 A164707 A164708

Keyword

nonn,tabf

Author

Geoffrey Critzer, Aug 23 2009

Status

approved