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A339565
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Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).
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2
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1, 3, 17, 101, 627, 3999, 25955, 170571, 1131433, 7559301, 50795985, 342935689, 2324278669, 15804931797, 107775401349, 736723618773, 5046774983235, 34636814325087, 238114193665451, 1639378334244867, 11301978856210543, 78010917772099207, 539055832175992119
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [(x*y)^n] 1/(1-x-y-x*y-x*y^2-x^2*y). - Alois P. Heinz, Dec 09 2020
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MAPLE
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a:= proc(n) local t; 1/(1-x-y-x*y-(x*y^2)-(x^2*y));
for t in [x, y] do coeftayl(%, t=0, n) od
end:
# second Maple program:
b:= proc(l) option remember; `if`(l[2]=0, 1,
add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
[[1, 0], [0, 1], [1$2], [1, 2], [2, 1]]))
end:
a:= n-> b([n$2]):
# third Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 3, 17][n+1],
((6*n-3)*a(n-1)+(7*n-7)*a(n-2)+(4*n-6)*a(n-3))/n)
end:
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MATHEMATICA
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b[l_] := b[l] = If[l[[2]] == 0, 1,
Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l - h]], {h,
{{1, 0}, {0, 1}, {1, 1}, {1, 2}, {2, 1}}}]];
a[n_] := b[{n, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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