A292436 Array T read by antidiagonals: T(m,n) is the number of lattice walks of minimal length from (0,0) to (m,n) using steps in directions from {(1,0), (0,1), (2,1), (1,2)}.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 9, 3, 1, 1, 4, 1, 2, 1, 4, 1, 1, 5, 3, 9, 9, 3, 5, 1, 1, 6, 6, 24, 36, 24, 6, 6, 1, 1, 7, 10, 1, 3, 3, 1, 10, 7, 1, 1, 8, 15, 4, 16, 24, 16, 4, 15, 8, 1, 1, 9, 21, 10, 50, 100, 100, 50, 10, 21, 9, 1, 1, 10, 28, 20, 1, 4, 6, 4, 1, 20, 28, 10, 1

Offset: 0

Author

Steven Klee, Dec 08 2017

Keywords

• nonn
• walk
• tabl

Examples

			Array T(m,n) begins
n\m 0    1    2    3    4    5    6    7    8    9   10
0   1    1    1    1    1    1    1    1    1    1    1
1   1    2    1    2    3    4    5    6    7    8    9
2   1    1    4    9    1    3    6   10   15   21   28
3   1    2    9    2    9   24    1    4   10   20   35
4   1    3    1    9   36    3   16   50    1    5   15
5   1    4    3   24    3   24  100    4   25   90    1
6   1    5    6    1   16  100    6   50  225    5   36
7   1    6   10    4   50    4   50  300   10   90  441
8   1    7   15   10    1   25  225   10  120  735   15
9   1    8   21   20    5   90    5   90  735   20  245
10  1    9   28   35   15    1   36  441   15  245 1960
		

Links

• Jackson Evoniuk, Steven Klee, Van Magnan, Enumerating Minimal Length Lattice Paths, 2017, also Enumerating Minimal Length Lattice Paths, J. Int. Seq., Vol. 21 (2018), Article 18.3.6.

Crossrefs

Cf. A007318, A292435.

Programs

Sage

# For an implementation see A292435.

Formula

T(m,n) = binomial(m-n,n) for m>=2*n;

T(m,n) = binomial(q+r,r)*binomial(q+r,m-q) for 1/2*n<=m<=2*n, where m+n = 3*q+r with 0<=r<=2;

T(m,n) = binomial(n-m,m) for m<=1/2*n.