cp's OEIS Frontend

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.

%I A047089 #17 Aug 10 2024 01:30:25
%S A047089 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,
%T A047089 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,
%U A047089 139,139,98,64,34,9,1,1,10,46,126,228,278,237,98,98,43,10,1,1,11,56,172
%N 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.
%C A047089 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).
%C A047089 "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.
%C A047089 "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).
%C A047089 "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."
%e A047089 Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1};...
%o A047089 (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 */
%o A047089 (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 */
%Y A047089 See also the related array A107027.
%K A047089 nonn,tabl
%O A047089 0,8
%A A047089 _Clark Kimberling_
%E A047089 "Diagonals" in definition changed to "antidiagonals" by _Michael Somos_, Aug 19 2007