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A198324
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Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1) and S=(0,1).
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4
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1, 0, 1, 1, 4, 10, 35, 116, 427, 1584, 6146, 24216, 97754, 400080, 1662645, 6986127, 29669872, 127101015, 548839687, 2386211664, 10439207266, 45920497075, 203004397362, 901459381683, 4019351034816, 17987665701788, 80773320086286, 363842478143834
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OFFSET
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0,5
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) ~ c * (2*(1+sqrt(2)))^n / n^(3/2), where c = 0.01202323187423280845930143205554758... . - Vaclav Kotesovec, Sep 07 2014
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EXAMPLE
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a(4) = 4: UDSDSD, UDUD, UDSSDD, UUDD.
a(5) = 10: UDSDSDSD, UDUDSD, UDSSDDSD, UUDDSD, UDSDUD, UDSDSSDD, UDUSDD, UDSSDSDD, UUDSDD, UDSUDD.
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MAPLE
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b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
`if`(y>0, b(x, y-1)+b(x-1, y-1), 0)+b(x-1, y+1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);
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MATHEMATICA
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b[x_, y_] := b[x, y] = If[y>x, 0, If[x == 0, 1, If[y>0, b[x, y-1] + b[x-1, y-1], 0] + b[x-1, y+1]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A000108 (without S-steps), A224769 (paths to (n,n)), A225041 (with additional H-steps), A286427.
Sequence in context: A030003 A339845 A234009 * A149175 A149176 A059710
Adjacent sequences: A198321 A198322 A198323 * A198325 A198326 A198327
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Apr 18 2013
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STATUS
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approved
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