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.