A333098 -id:A333098 - cp's OEIS frontend

A330169 a(n) is the total area of all closed Deutsch paths of length n.

1, 3, 12, 39, 129, 411, 1300, 4065, 12633, 39046, 120204, 368844, 1128837, 3447303, 10508592, 31985085, 97226733, 295214316, 895502520, 2714106318, 8219809425, 24877611798, 75248738292, 227488953354, 687408882709, 2076269682831, 6268788729240, 18920387069731, 57086882549253

Offset: 2

Michel Marcus, Mar 05 2020

nonn

Deutsch paths are a variation of Dyck paths that allow for down-steps of arbitrary length.

Alois P. Heinz, Table of n, a(n) for n = 2..2096
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8.

Cf. A001006 (Motzkin numbers), A005043, A333017, A333098.

a:= proc(n) option remember;`if`(n<4, [0$2, 1, 3][n+1], (4*n*
      a(n-1)+(2*n+4)*a(n-2)+12*(1-n)*a(n-3)+9*(1-n)*a(n-4))/(n+1))
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Mar 05 2020

a = DifferenceRoot[Function[{y, n}, {9(n+3)y[n] + 12(n+3)y[n+1] - 2(n+6)y[n+2] - 4(n+4)y[n+3] + (n+5)y[n+4] == 0, y[2] == 1, y[3] == 3, y[4] == 12, y[5] == 39}]];
a /@ Range[2, 30] (* Jean-François Alcover, Mar 12 2020 *)

my(z='z+O('z^30), v=(1-z-sqrt(1-2*z-3*z^2))/(2*z)); Vec(v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2))

G.f.: v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2) where v=(1-z-sqrt(1-2*z-3*z^2))/(2*z), that is, where v is the g.f. of A001006.

A333114 Sum over all closed Deutsch paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 0, 1, 1, 5, 11, 44, 134, 529, 1902, 7793, 31068, 133641, 574259, 2594969, 11842726, 56083004, 269450143, 1333170844, 6703500545, 34548749471, 181026885253, 969167994094, 5273977173249, 29257773480987, 164894374634333, 945779302210358, 5507572390808676

Offset: 0

Alois P. Heinz, Mar 07 2020

nonn

Deutsch paths (named after their inventor Emeric Deutsch by Helmut Prodinger) are like Dyck paths where down steps can get to all lower levels. Open paths can end at any level, whereas closed paths have to return to the lowest level zero at the end.

			a(4) = (1/1)*(3/1) + 2/2 + 3/3 = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..800
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
Wikipedia, Counting lattice paths

Cf. A005043, A005411, A330169, A333098.

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      `if`(t and j<0, x/y, 1)*b(x-1, y+j, is(j>0)), j=[
      `if`(y=0, [][], -1), $1..x-1-y]))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..30);

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[If[t && j < 0, x/y, 1]* b[x-1, y+j, j > 0], {j, Join[If[y == 0, {}, {-1}], Range[x-1-y]]}]];
a[n_] := b[n, 0, False];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 19 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A330169 a(n) is the total area of all closed Deutsch paths of length n.

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