A229756 Triangle T(n,k): the number of binary sequences of n zeros and n ones in which the longest run is of length k.

2, 2, 4, 2, 12, 6, 2, 32, 28, 8, 2, 82, 110, 48, 10, 2, 206, 408, 224, 72, 12, 2, 516, 1454, 968, 378, 100, 14, 2, 1294, 5048, 4016, 1784, 578, 132, 16, 2, 3252, 17244, 16202, 7980, 2924, 830, 168, 18, 2, 8194, 58290, 64058, 34570, 13810, 4464, 1140, 208, 20

Offset

1,1

Comments

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

Links

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

Formula

Let h(n,p,k) := sum(j=0..floor((n-p)/k), (-1)^j*C(p,j)*C(n-1-j*k,p-1)) with h(n,p,0) := 0, and let g(n,k) := 2*sum(i=1..n, h(n,i,k)*(h(n,i,k)+h(n,i+1,k))). Then T(n,k) = g(n,k)-g(n,k-1).

Example

The triangle begins:
2
2  4
2 12   6
2 32  28  8
2 82 110 48 10
The second row counts the sets {0101, 1010} and {0011, 0110, 1001, 1100}.

Prog

(PARI)
h(n, p, k)=if(k==0, 0, sum(j=0, floor((n-p)/k), (-1)^j*binomial(p, j)*binomial(n-1-j*k, p-1)))
g(n, k)=2*sum(i=1, n, h(n, i, k)*(h(n, i, k)+h(n, i+1, k)))
T(n, k)=g(n, k)-g(n, k-1)
r(n)=vector(n, x, 2*T(n, x))

Crossrefs

Sequence in context: A067228 A356543 A332002 * A227450 A010026 A059427

Adjacent sequences: A229753 A229754 A229755 * A229757 A229758 A229759

Keyword

nonn,tabl

Author

Andrew Woods, Sep 28 2013

Status

approved