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A225042
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Number of lattice paths from (0,0) to (n,n) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).
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6
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1, 2, 8, 48, 360, 3088, 28928, 288208, 3003952, 32402384, 359019952, 4064452272, 46829600704, 547498996736, 6480275672192, 77511461858592, 935562094075392, 11381614588917296, 139425068741674448, 1718444636265140992, 21295889048851102176, 265200380258393530896
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OFFSET
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0,2
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..890
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.56165398271839628518..., c = 0.03237684690282108810066870410351693504744... . - Vaclav Kotesovec, Sep 07 2014
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EXAMPLE
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a(0) = 1: the empty path.
a(1) = 2: U, HS.
a(2) = 8: UU, HSU, UHS, HSHS, HUS, HHSS, UDSS, HSDSS.
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MAPLE
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b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
b(x-1, y)+`if`(y>0, b(x-1, y-1)+b(x, y-1), 0)+b(x-1, y+1)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..25);
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MATHEMATICA
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b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, b[x - 1, y] + If[y > 0, b[x - 1, y - 1] + b[x, y - 1], 0] + b[x - 1, y + 1]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)
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CROSSREFS
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Cf. A006318 (without D-steps), A224769 (without H-steps), A224776 (without U-steps), A225041 (paths to (n,0)), A286765.
Sequence in context: A085615 A054726 A003576 * A326887 A095989 A177388
Adjacent sequences: A225039 A225040 A225041 * A225043 A225044 A225045
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Apr 25 2013
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STATUS
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approved
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