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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 (list; table; graph; refs; listen; history; text; internal format)
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."

LINKS

Table of n, a(n) for n=0..81.

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

Adjacent sequences: A047086 A047087 A047088 * A047090 A047091 A047092

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

EXTENSIONS

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

STATUS

approved

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Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)