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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)}.
0
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
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
Sequence in context: A175466 A214403 A261527 * A184097 A345932 A205399
KEYWORD
nonn,walk,tabl
AUTHOR
Steven Klee, Dec 08 2017
STATUS
approved