A047089 Array T read by antidiagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and touches the line y=x/2 only at lattice points.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 7, 4, 4, 1, 1, 5, 11, 11, 8, 5, 1, 1, 6, 16, 22, 19, 13, 6, 1, 1, 7, 22, 38, 41, 19, 19, 7, 1, 1, 8, 29, 60, 79, 60, 38, 26, 8, 1, 1, 9, 37, 89, 139, 139, 98, 64, 34, 9, 1, 1, 10, 46, 126, 228, 278, 237, 98, 98, 43, 10, 1, 1, 11, 56, 172

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

Comments from Timothy Y. Chow (tchow(AT)alum.mit.edu), Nov 15 2006 on this sequence and A107027. "If you replace "the line y = x/2" with "the line y = x/(n-1)" in the definition of this sequence, then the formula for T(h,k) becomes (h+k choose k) - (n-1)*(h+k choose k-1).

"As for A107027, it has a combinatorial interpretation: T(n,k) is the number of paths of length n*k such that each step has length 1 directed up or right and touches the line y = x/(n-1) only at lattice points.

"To see this, let us avoid notational confusion by replacing the "k" in A047089 by "j". Then the formula above becomes (h+j choose j) - (n-1)*(h+j choose j-1).

"If we sum over all the points at a distance n*k from (0,0) - i.e. if we sum from j=0 to j=k and let h = n*k-j - then we get (n*k choose k) - (n-2) * sum_{j=0}^{k-1} (n*k choose j) This is equivalent to the formula you report for A107027."

Examples

			Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1};...
		

Crossrefs

See also the related array A107027.

Programs

PARI

{T(n, k) = local(v); if( k<0 || k>n, 0, for(i=1, n+1, v=vector(i, j, if( j<2 || j>i-1, 1, v[j-1] + if( i%3 || i!=j+i\3, v[j])))); v[k+1])}; /* Michael Somos, Jan 28 2004 */

PARI

{T(n, k) = if( k<0 || k>n, 0, if( n==0 && k==0, 1, T(n-1, k-1) + if( (n+1)%3 || n!=k+(n+1)\3, T(n-1, k))))}; /* Michael Somos, Jan 28 2004 */

Extensions

"Diagonals" in definition changed to "antidiagonals" by Michael Somos, Aug 19 2007