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.
Original entry on oeis.org
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
- 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.
-
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 *)
A276901
Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 2, and staying strictly above the x-axis.
Original entry on oeis.org
0, 1, 2, 9, 34, 159, 730, 3579, 17762, 90538, 467796, 2452727, 12997554, 69549847, 375159290, 2038068813, 11140256754, 61227097438, 338140106124, 1875581756078, 10444142352812, 58364192607047, 327203347219250, 1839778650617309, 10372512509521074
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1291
- 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.
-
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 *)
A276901[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 2, 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
- 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.
-
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
- 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.
-
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 *)
A278398
Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-3,-2,-1,1,2,3}.
Original entry on oeis.org
1, 3, 15, 75, 400, 2169, 11989, 66985, 377718, 2144290, 12240943, 70193305, 404029950, 2332989921, 13508237399, 78399357623, 455959701700, 2656652705422, 15504203678738, 90614205677898, 530288460288008, 3107012752773125, 18223934202102463, 106996319699099591
Offset: 0
- Andrew Howroyd, 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:1609.06473 [math.CO], 2016.
-
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[24] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)
-
seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i)-1, p=1); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018
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
- 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.
-
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
- 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.
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
- 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.
-
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
- 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.
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