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A284778
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Number of self-avoiding planar walks of length n+1 starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.
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10
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0, 1, 1, 4, 8, 22, 54, 142, 370, 983, 2627, 7086, 19238, 52561, 144377, 398518, 1104794, 3074809, 8588093, 24064642, 67630898, 190584766, 538412426, 1524554956, 4326119748, 12300296227, 35037658099, 99977847308, 285741659312, 817901027070, 2344475178110
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OFFSET
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0,4
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COMMENTS
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From Gus Wiseman, Nov 15 2022: (Start)
Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 3 vertices and more than one branch of the root. This would imply a(n) = A187306(n+1) - A005043(n+1). For example, the a(1) = 1 through a(5) = 22 trees are:
(ooo) (oooo) (ooooo) (oooooo) (ooooooo)
((oo)oo) ((oo)ooo) ((oo)oooo)
(o(oo)o) ((ooo)oo) ((ooo)ooo)
(oo(oo)) (o(oo)oo) ((oooo)oo)
(o(ooo)o) (o(oo)ooo)
(oo(oo)o) (o(ooo)oo)
(oo(ooo)) (o(oooo)o)
(ooo(oo)) (oo(oo)oo)
(oo(ooo)o)
(oo(oooo))
(ooo(oo)o)
(ooo(ooo))
(oooo(oo))
(((oo)o)oo)
((o(oo))oo)
((oo)(oo)o)
((oo)o(oo))
(o((oo)o)o)
(o(o(oo))o)
(o(oo)(oo))
(oo((oo)o))
(oo(o(oo)))
(End)
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..2105
Alois P. Heinz, Animation of a(6)=54 walks
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FORMULA
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G.f.: (1-2*x-x^2-sqrt(1-4*x+2*x^2+4*x^3-3*x^4))/(2*(x+1)*x^3).
Recursion: see Maple program.
a(n) = A284414(n,n+1) = A284652(n,n+1).
a(n) ~ 3^(n+5/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 02 2017
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MAPLE
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a:= proc(n) option remember; `if`(n<3, (3-n)*n/2,
((n^2-n+3)*a(n-1)+(5*n-2)*n*a(n-2)+
3*(n-1)*n*a(n-3))/((n+3)*(n-1)))
end:
seq(a(n), n=0..35);
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MATHEMATICA
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CoefficientList[Series[(1 - 2*x - x^2 - Sqrt[1 - 4*x + 2*x^2 + 4*x^3 - 3*x^4])/(2*(x + 1)*x^3), {x, 0, 50}], x] (* Indranil Ghosh, Apr 02 2017 *)
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CROSSREFS
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First upper diagonal of A284414, A284652.
CF. A005043, A187306.
Sequence in context: A297339 A290138 A266922 * A057583 A129788 A170938
Adjacent sequences: A284775 A284776 A284777 * A284779 A284780 A284781
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KEYWORD
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nonn,walk
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AUTHOR
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Alois P. Heinz, Apr 02 2017
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STATUS
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approved
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