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EXAMPLE
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a(2) = 8: [(0,0),(1,0),(1,1),(2,1),(2,2)], [(0,0),(1,0),(1,1),(2,2)], [(0,0),(1,0),(2,0),(2,1),(2,2)], [(0,0),(1,0),(2,1),(2,2)], [(0,0),(1,0),(2,2)], [(0,0),(1,1),(2,1),(2,2)], [(0,0),(1,1),(2,2)], [(0,0),(2,1),(2,2)].
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MAPLE
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b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y) -b(x-1, y-1)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..30);
# second Maple program gives series:
series(RootOf(x^4*T^4-(x^2+1)*x^2*T^3-(x^2-2*x-2)*x*T^2-(x^2+1)*T+1, T), x=0, 31); # Mark van Hoeij, Apr 17 2013
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