A276901 -id:A276901 - cp's OEIS frontend

A276852 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

0, 1, 2, 7, 28, 121, 560, 2677, 13230, 66742, 343092, 1788681, 9439870, 50321865, 270594896, 1465941763, 7993664588, 43839212778, 241650560756, 1338084935826, 7439615051328, 41516113036777, 232452845782308, 1305500166481715, 7352433083806020, 41514430735834714

Offset: 0

Michael Wallner, Sep 21 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1292
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A276901, A276902, A276903, A276904.

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k - x, h], {x, h}] + Sum[walks[n - 1, k + x, h], {x, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... -1, 1, ..., h} that end at altitude k *)
A276852[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 1, 3]) (* Davin Park, Oct 10 2016 *)

A276902 Number of positive walks with n steps {-3,-2,-1,0,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 12, 56, 284, 1526, 8530, 49106, 289149, 1733347, 10542987, 64904203, 403632551, 2531971729, 16002136283, 101795589297, 651286316903, 4188174878517, 27055199929042, 175488689467350, 1142479579205721, 7462785088260791, 48896570201100002

Offset: 0

Michael Wallner, Sep 21 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1189
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A276852, A276901, A276903, A276904.

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k + x, h], {x, -h, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... , h} that end at altitude k *)
A276902[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 1, 3]) (* Davin Park, Oct 10 2016 *)

A276903 Number of positive walks with n steps {-2,-1,0,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 2, 7, 25, 96, 382, 1567, 6575, 28096, 121847, 534953, 2373032, 10619922, 47890013, 217395690, 992640367, 4555957948, 21007405327, 97266928685, 452046424465, 2108022305795, 9860773604035, 46256877824220, 217555982625385, 1025667805621986, 4846240583558277

Offset: 0

Michael Wallner, Sep 21 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1437
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A276852, A276901, A276902, A276904.

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k + x, h], {x, -h, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... , h} that end at altitude k *)
A276903[n_] := (Do[walks[m, k, 2], {m, n}, {k, 2 m}]; walks[n, 2, 2]) (* Davin Park, Oct 10 2016 *)

A276904 Number of positive walks with n steps {-3,-2,-1,0,1,2,3} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 14, 68, 358, 1966, 11172, 65104, 387029, 2337919, 14309783, 88555917, 553171371, 3483277785, 22087378303, 140913963221, 903876307075, 5825742149049, 37710582868464, 245052827645474, 1598017940728401, 10454217006683855, 68591382498826168

Offset: 0

Michael Wallner, Sep 21 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1189
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A276852, A276901, A276902, A276903.

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k + x, h], {x, -h, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... , h} that end at altitude k *)
A276904[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 2, 3]) (* Davin Park, Oct 10 2016 *)

More terms from Alois P. Heinz, Oct 10 2016

A111160 G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0.

Original entry on oeis.org

0, 1, 1, 4, 9, 31, 91, 309, 1009, 3481, 11956, 42065, 148655, 532039, 1915369, 6950452, 25357233, 93034813, 342888250, 1269246437, 4715945712, 17583623988, 65766726906, 246694006971, 927801717255, 3497918129001, 13217196871126, 50046561077947

Offset: 0

N. J. A. Sloane, Oct 22 2005

nonn

Expressible in terms of ballot numbers.

Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016

T. D. Noe, Table of n, a(n) for n=0..200
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
D. G. Rogers, Comments on A111160, A055113 and A006013

Cf. A000108, A055113, A006013, A187430, A276901, A306668.

I:=[1,1,4]; [0] cat [n le 3 select I[n] else (n*(115*n^3 - 344*n^2 + 299*n - 82)*Self(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*Self(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*Self(n-3))/(2*n*(n+1)*(2*n+1)*(5*n-8)): n in [1..30]]; // Vincenzo Librandi, Oct 06 2015

a := n -> (-1)^(n+1)*binomial(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/ (2*n+1);
[0, op([seq(round(evalf(a(n),32)), n=1..27)])]; # Peter Luschny, Oct 06 2015

CoefficientList[ Series[ -((-3 + Sqrt[1 - 4*x] + Sqrt[2]*Sqrt[1 + Sqrt[1 - 4x] + 6x])/(4x)), {x, 0, 10}], x] (* Robert G. Wilson v *)

a(n) = if(n==0, 0, sum(k=0, (n+1)/2, binomial(n-k,n-2*k+1)*binomial(2*n+1,k))/(2*n+1)); \\ Altug Alkan, Oct 05 2015

Let C := (1 - sqrt(1 - 4*x)) / (2*x), Z := (- 1/4 - (1/4)*(1 - 4*x)^(1/2) + (1/4)*(2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2))/x; g.f. is W := C - Z.
G.f.: -((-3 + sqrt(1 - 4x) + sqrt(2)*sqrt(1 + sqrt(1 - 4x) + 6x))/(4x)).
a(n) = sum(j=0..n+1, binomial(n+2*j-1,j)*(-1)^(n+j+1)*binomial(2*n+1,j+n))/(2*n+1). [Vladimir Kruchinin, Feb 15 2013]
a(n) ~ (1+1/sqrt(5))*2^(2*n-1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: 2*n*(n+1)*(2*n+1)*(5*n-8)*a(n) = n*(115*n^3 - 344*n^2 + 299*n - 82)*a(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*a(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) = Sum_{j=0..(n+1)/2}(binomial(n-j,n-2*j+1)*binomial(2*n+1,j))/(2*n+1). - Vladimir Kruchinin, Oct 05 2015
a(n) = (-1)^(n+1)*C(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/(2*n+1) for n>0. - Peter Luschny, Oct 06 2015

A277920 Number of positive walks with n steps {-4,-3,-2,-1,0,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 4, 20, 120, 780, 5382, 38638, 285762, 2162033, 16655167, 130193037, 1030117023, 8234025705, 66391916397, 539360587341, 4410492096741, 36274113675369, 299864297741292, 2490192142719336, 20764402240048267, 173784940354460219, 1459360304511145146

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1054
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A276902, A276852, A276901, A276903, A276904.

b:= proc(n, y) option remember; `if`(n=0, `if`(y=1, 1, 0),
      add((h-> `if`(h<1, 0, b(n-1, h)))(y+d), d=-4..4))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 12 2016

b[n_, y_] := b[n, y] = If[n == 0, If[y == 1, 1, 0], Sum[Function[h, If[h < 1, 0, b[n - 1, h]]][y + d], {d, -4, 4}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 03 2017, after Alois P. Heinz *)

A277921 Number of positive walks with n steps {-4,-3,-2,-1,0,1,2,3,4} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 4, 23, 142, 950, 6662, 48420, 361378, 2753687, 21334313, 167551836, 1330894754, 10673486660, 86306300366, 702872359332, 5759986152740, 47463395965108, 393027545388119, 3268814565684836, 27294209365111429, 228718165320327356, 1922825557218427271

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1053
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A276904, A276852, A276901, A276902, A276903.

A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 13, 71, 405, 2501, 15923, 104825, 704818, 4827957, 33549389, 235990887, 1676907903, 12019875907, 86804930199, 630999932585, 4613307289260, 33900874009698, 250257489686870, 1854982039556397, 13800559463237465, 103017222722691145, 771348369563479705

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A276852, A276901, A276902, A276903, A276904.

b:= proc(n, y) option remember; `if`(n=0, `if`(y=1, 1, 0), add
     ((h-> `if`(h<1, 0, b(n-1, h)))(y+d), d=[$-4..-1, $1..4]))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 12 2016

b[n_, y_] := b[n, y] = If[n == 0, If[y == 1, 1, 0], Sum[Function[h, If[h < 1, 0, b[n - 1, h]]][y + d], {d, Join[Range[-4, -1], Range[4]]}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 03 2017, after Alois P. Heinz *)

A277923 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 3, 16, 84, 505, 3121, 20180, 133604, 904512, 6224305, 43432093, 306524670, 2184389874, 15695947669, 113595885023, 827299204132, 6058526521135, 44586954104578, 329579179316696, 2445858862779018, 18216235711289695, 136113075865844577, 1020074492384232296

Offset: 0

David Nguyen, Nov 04 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1113
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A276901, A276852, A276902, A276903, A276904.`

cp's OEIS Frontend

A276852 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

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A276902 Number of positive walks with n steps {-3,-2,-1,0,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

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A276903 Number of positive walks with n steps {-2,-1,0,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

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A276904 Number of positive walks with n steps {-3,-2,-1,0,1,2,3} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

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A111160 G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0.

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A277920 Number of positive walks with n steps {-4,-3,-2,-1,0,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

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A277921 Number of positive walks with n steps {-4,-3,-2,-1,0,1,2,3,4} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

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A277922 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

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A277923 Number of positive walks with n steps {-4,-3,-2,-1,1,2,3,4} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.

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