A227924 - OEIS

%I #8 Oct 11 2013 21:23:58

%S 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,

%T 12662,182,20,4,0,0,0,2,48102,466,38,12,0,0,0,0,2,183460,1186,84,20,4,

%U 0,0,0,0,2

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

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

%H Andrew Woods, <a href="/A227924/b227924.txt">Rows n = 1..50 of triangle, flattened</a>

%e The triangle begins:

%e [2,

%e [4,   2,

%e [18,  0,  2,

%e [64,  4,  0, 2,

%e [238, 12, 0, 0, 2,

%e The second row counts the sets {0101, 1010, 0110, 1001} and {0011, 1100}.

%o (PARI)

%o bn(n,k)=binomial(max(0,n),k)

%o 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)))

%o T(n,k)=f(n,k)-f(n,k+1)

%o r(n)=vector(n,x,T(n,x))

%Y Cf. A229756.

%K nonn,tabl

%O 1,1

%A _Andrew Woods_, Oct 09 2013