A224769 -id:A224769 - cp's OEIS frontend

A225042 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).

1, 2, 8, 48, 360, 3088, 28928, 288208, 3003952, 32402384, 359019952, 4064452272, 46829600704, 547498996736, 6480275672192, 77511461858592, 935562094075392, 11381614588917296, 139425068741674448, 1718444636265140992, 21295889048851102176, 265200380258393530896

Offset: 0

Alois P. Heinz, Apr 25 2013

nonn

			a(0) = 1: the empty path.
a(1) = 2: U, HS.
a(2) = 8: UU, HSU, UHS, HSHS, HUS, HHSS, UDSS, HSDSS.

Alois P. Heinz, Table of n, a(n) for n = 0..890

Cf. A006318 (without D-steps), A224769 (without H-steps), A224776 (without U-steps), A225041 (paths to (n,0)), A286765.

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);

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 *)

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 = 0.03237684690282108810066870410351693504744294274892020985727414558915214336... - Vaclav Kotesovec, Sep 07 2014, updated Sep 13 2021

A198324 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).

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Apr 18 2013

nonn

			a(4) = 4: UDSDSD, UDUD, UDSSDD, UUDD.
a(5) = 10: UDSDSDSD, UDUDSD, UDSSDDSD, UUDDSD, UDSDUD, UDSDSSDD, UDUSDD, UDSSDSDD, UUDSDD, UDSUDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108 (without S-steps), A224769 (paths to (n,n)), A225041 (with additional H-steps), A286427.

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);

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 *)

a(n) ~ c * (2*(1+sqrt(2)))^n / n^(3/2), where c = 0.01202323187423280845930143205554758... . - Vaclav Kotesovec, Sep 07 2014

A286425 Total number of nodes summed over all 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) and S=(0,1).

Original entry on oeis.org

1, 2, 8, 44, 285, 2028, 15338, 120960, 983108, 8172094, 69116592, 592590616, 5136777504, 44928712804, 395907022448, 3510622573064, 31296093794827, 280275392413204, 2520017580255461, 22736733105613548, 205767848345966976, 1867240544055742660

Offset: 0

Alois P. Heinz, May 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A224769.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
      (p-> p+[0, p[1]])(b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..30);

a(n) ~ c * d^n / sqrt(n), where d = (3*(71 + 8*sqrt(2))^(1/3))/4 + 51/(4*(71 + 8*sqrt(2))^(1/3)) + 13/4 = 9.443535601593252082001105527294087383986236797... and c = 0.0201623254316291127574085659620180015446126055020315052104102916... - Vaclav Kotesovec, Sep 11 2021

cp's OEIS Frontend

A225042 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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A198324 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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A286425 Total number of nodes summed over all 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) and S=(0,1).

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Author

Keywords

Links

Crossrefs

Programs

Maple

Formula