A164705 T(n,k) = binomial(2n-k,n) * 2^(k-1), T(0,0)=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 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, 256

Offset: 0

Author

Geoffrey Critzer, Aug 23 2009

Keywords

• nonn
• tabl

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) = A088218(n).

Examples

			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,   1;
   3,   3,   2;
  10,  10,   8,  4;
  35,  35,  30, 20,  8;
 126, 126, 112, 84, 48, 16;
 ...
		

Links

• Sean A. Irvine, Computing A382782, 2025.

Crossrefs

Row sums give A081294.

Main diagonal gives A011782.

Cf. A000531, A088218.

Programs

Maple

T:= (n, k)-> ceil(binomial(2*n-k, n)*2^(k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 06 2025

Mathematica

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

Formula

Sum_{k=0..n} k * T(n,k) = A000531(n). - Alois P. Heinz, Apr 06 2025

Extensions

T(0,0)=1 prepended by Sean A. Irvine, Apr 05 2025