A330601 Array T read by antidiagonals: T(m,n) is the number of lattice walks from (0,0) to (m,n) using one step from {(3,0), (2,1), (1,2), (0,3)} and all other steps from {(1,0), (0,1)}.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, 2, 3, 9, 12, 12, 9, 3, 4, 16, 28, 32, 28, 16, 4, 5, 25, 55, 75, 75, 55, 25, 5, 6, 36, 96, 156, 180, 156, 96, 36, 6, 7, 49, 154, 294, 392, 392, 294, 154, 49, 7, 8, 64, 232, 512, 784, 896, 784, 512, 232, 64, 8, 9, 81, 333, 837, 1458, 1890, 1890, 1458, 837, 333, 81, 9

Offset: 0

Author

Steven Klee, Dec 19 2019

Keywords

• tabl
• nonn

Examples

			For (m,n) = (3,1), there are T(3,1) = 4 paths:
(3,0), (0,1)
(0,1), (3,0)
(2,1), (1,0)
(1,0), (2,1).
Array T(m,n) begins
n/m 0   1    2     3     4      5      6      7       8       9
0   0   0    0     1     2      3      4      5       6       7
1   0   0    1     4     9     16     25     36      49      64
2   0   1    4    12    28     55     96    154     232     333
3   1   4   12    32    75    156    294    512     837    1300
4   2   9   28    75   180    392    784   1458    2550    4235
5   3  16   55   156   392    896   1890   3720    6897   12144
6   4  25   96   294   784   1890   4200   8712   17028   31603
7   5  36  154   512  1458   3720   8712  19008   39039   76076
8   6  49  232   837  2550   6897  17028  39039   84084  171600
9   7  64  333  1300  4235  12144  31603  76076  171600  366080
		

Crossrefs

T(m,0) is A000027 for m >= 2.

T(m,1) is A000290 for m >= 1.

T(m,2) is A006000.

Programs

Sage

def T(m,n):
    return (m+n-2)*(binomial(m+n-2, m) + binomial(m+n-2, n))

Formula

T(m,n) = (m+n-2)*(binomial(m+n-2,m) + binomial(m+n-2,n)).