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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.
12
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
0,8
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."
EXAMPLE
Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1};...
PROG
(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 */
CROSSREFS
See also the related array A107027.
Sequence in context: A104769 A078013 A086461 * A122218 A072405 A146565
KEYWORD
nonn,tabl
EXTENSIONS
"Diagonals" in definition changed to "antidiagonals" by Michael Somos, Aug 19 2007
STATUS
approved