A383552 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,1),(2,2).
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 12, 12, 5, 1, 1, 6, 18, 26, 18, 6, 1, 1, 7, 25, 47, 47, 25, 7, 1, 1, 8, 33, 76, 101, 76, 33, 8, 1, 1, 9, 42, 114, 189, 189, 114, 42, 9, 1, 1, 10, 52, 162, 321, 404, 321, 162, 52, 10, 1, 1, 11, 63, 221, 508, 772, 772, 508, 221, 63, 11, 1
Offset: 0
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, ... 1, 3, 7, 12, 18, 25, 33, ... 1, 4, 12, 26, 47, 76, 114, ... 1, 5, 18, 47, 101, 189, 321, ... 1, 6, 25, 76, 189, 404, 772, ... 1, 7, 33, 114, 321, 772, 1645, ...
Programs
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PARI
a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x-y-x^2*y^2), n), k);
Formula
A(n,k) = A(k,n).
A(n,k) = A(n-1,k) + A(n,k-1) + A(n-2,k-2).
G.f.: 1 / (1 - x - y - x^2*y^2).