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%I #10 Jan 10 2020 13:18:12
%S 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,
%T 55,75,75,55,25,5,6,36,96,156,180,156,96,36,6,7,49,154,294,392,392,
%U 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
%N 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)}.
%F T(m,n) = (m+n-2)*(binomial(m+n-2,m) + binomial(m+n-2,n)).
%e For (m,n) = (3,1), there are T(3,1) = 4 paths:
%e (3,0), (0,1)
%e (0,1), (3,0)
%e (2,1), (1,0)
%e (1,0), (2,1).
%e Array T(m,n) begins
%e n/m 0 1 2 3 4 5 6 7 8 9
%e 0 0 0 0 1 2 3 4 5 6 7
%e 1 0 0 1 4 9 16 25 36 49 64
%e 2 0 1 4 12 28 55 96 154 232 333
%e 3 1 4 12 32 75 156 294 512 837 1300
%e 4 2 9 28 75 180 392 784 1458 2550 4235
%e 5 3 16 55 156 392 896 1890 3720 6897 12144
%e 6 4 25 96 294 784 1890 4200 8712 17028 31603
%e 7 5 36 154 512 1458 3720 8712 19008 39039 76076
%e 8 6 49 232 837 2550 6897 17028 39039 84084 171600
%e 9 7 64 333 1300 4235 12144 31603 76076 171600 366080
%o (Sage)
%o def T(m,n):
%o return (m+n-2)*(binomial(m+n-2, m) + binomial(m+n-2, n))
%Y T(m,0) is A000027 for m >= 2.
%Y T(m,1) is A000290 for m >= 1.
%Y T(m,2) is A006000.
%K tabl,nonn
%O 0,11
%A _Steven Klee_, Dec 19 2019
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