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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, 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
0,11
FORMULA
T(m,n) = (m+n-2)*(binomial(m+n-2,m) + binomial(m+n-2,n)).
EXAMPLE
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
PROG
(Sage)
def T(m, n):
return (m+n-2)*(binomial(m+n-2, m) + binomial(m+n-2, n))
CROSSREFS
T(m,0) is A000027 for m >= 2.
T(m,1) is A000290 for m >= 1.
T(m,2) is A006000.
Sequence in context: A097541 A151819 A079560 * A088680 A143194 A036264
KEYWORD
tabl,nonn
AUTHOR
Steven Klee, Dec 19 2019
STATUS
approved