A274742 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's that begin with 1 and have exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 3, 6, 6, 4, 1, 3, 9, 12, 8, 5, 1, 4, 12, 18, 20, 10, 6, 1, 4, 16, 30, 30, 30, 12, 7, 1, 5, 20, 40, 60, 45, 42, 14, 8, 1, 5, 25, 60, 80, 105, 63, 56, 16, 9, 1, 6, 30, 75, 140, 140, 168, 84, 72, 18, 10, 1, 6, 36, 105, 175, 280, 224, 252, 108, 90, 20, 11, 1, 7, 42, 126, 280, 350, 504, 336, 360, 135, 110, 22, 12, 1

Offset: 3

Author

Jeremy Dover, Jul 04 2016

Keywords

• nonn
• tabl

Comments

It appears that the row sums give the positive integers of A001629. - Omar E. Pol, Jul 09 2016

Examples

			n=3 => 100 -> T(3,0) = 1.
n=4 => 1001 -> T(4,0) = 1; 1100 -> T(4,1) = 1.
n=5 => 10010, 10100 -> T(5,0) = 1; 10011, 11001 -> T(5,1) = 2;
       11100 -> T(5,2) = 1.
Triangle starts:
1
1, 1
2, 2, 1
2, 4, 3, 1
3, 6, 6, 4, 1
3, 9, 12, 8, 5, 1
4, 12, 18, 20, 10, 6, 1
4, 16, 30, 30, 30, 12, 7, 1
5, 20, 40, 60, 45, 42, 14, 8, 1
5, 25, 60, 80, 105, 63, 56, 16, 9, 1
6, 30, 75, 140, 140, 168, 84, 72, 18, 10, 1
6, 36, 105, 175, 280, 224, 252, 108, 90, 20, 11, 1
7, 42, 126, 280, 350, 504, 336, 360, 135, 110, 22, 12, 1
		

Crossrefs

Cf. A046854, A274228.

Columns: A008619, A087811.

Programs

Mathematica

Table[Binomial[Floor[(n + k - 2)/2], k] Floor[(n - k - 1)/2], {n, 3, 15}, {k, 0, n - 3}] // Flatten (* Michael De Vlieger, Jul 05 2016 *)

PARI

t(n, k) = binomial(floor((n+k-2)/2), k) * floor((n-k-1)/2)
trianglerows(n) = for(x=3, n+2, for(y=0, x-3, print1(t(x, y), ", ")); print(""))
trianglerows(13) \\ Felix Fröhlich, Jul 05 2016

Formula

T(n,k) = binomial(floor((n+k-2)/2),k)*floor((n-k-1)/2).