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

0,5

Links

Table of n, a(n) for n=0..90.

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.

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.

Example

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

Prog

(Sage) # For an implementation see A292435.

Crossrefs

Cf. A007318, A292435.

Sequence in context: A175466 A214403 A261527 * A184097 A345932 A205399

Adjacent sequences: A292433 A292434 A292435 * A292437 A292438 A292439

Keyword

nonn,walk,tabl

Author

Steven Klee, Dec 08 2017

Status

approved