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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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: A318834 A067228 A332002 * A227450 A010026 A059427

Adjacent sequences:  A229753 A229754 A229755 * A229757 A229758 A229759

KEYWORD

nonn,tabl

AUTHOR

Andrew Woods, Sep 28 2013

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)