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
Examples
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}.
Links
- Andrew Woods, Rows n = 1..50 of triangle, flattened
Programs
-
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))
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).
Comments