A227924 Triangle T(n,k): the number of binary sequences of n zeros and n ones in which the shortest run is of length k.

2, 4, 2, 18, 0, 2, 64, 4, 0, 2, 238, 12, 0, 0, 2, 890, 28, 4, 0, 0, 2, 3348, 70, 12, 0, 0, 0, 2, 12662, 182, 20, 4, 0, 0, 0, 2, 48102, 466, 38, 12, 0, 0, 0, 0, 2, 183460, 1186, 84, 20, 4, 0, 0, 0, 0, 2

Offset: 1

Author

Andrew Woods, Oct 09 2013

Keywords

• nonn
• tabl

Comments

Row n sums to C(2*n,n) (A000984).

Examples

			The triangle begins:
  2,
  4,   2,
  18,  0,  2,
  64,  4,  0, 2,
  238, 12, 0, 0, 2,
  ...
The second row counts the sets {0101, 1010, 0110, 1001} and {0011, 1100}.
		

Links

• Andrew Woods, Rows n = 1..50 of triangle, flattened

Crossrefs

Cf. A229756.

Programs

PARI

bn(n,k)=binomial(max(0,n),k)
f(n,k)=2*sum(x=1,floor(n/k),bn(n+x*(1-k)-1,x-1)*(bn(n+x*(1-k)-1,x-1)+bn(n+(x+1)*(1-k)-1,x)))
T(n,k)=f(n,k)-f(n,k+1)
r(n)=vector(n,x,T(n,x))