A330169 -id:A330169 - cp's OEIS frontend

A333098 Number of closed Deutsch paths whose area is exactly n.

1, 1, 1, 2, 4, 6, 11, 21, 36, 64, 117, 208, 371, 669, 1197, 2141, 3844, 6888, 12336, 22119, 39644, 71034, 127323, 228200, 408955, 732957, 1313647, 2354298, 4219447, 7562249, 13553161, 24290307, 43533784, 78022169, 139833177, 250612596, 449153751, 804984038

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.

Alois P. Heinz, Table of n, a(n) for n = 0..1250
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
Wikipedia, Counting lattice paths

Cf. A330169.

b:= proc(x, y, k) option remember; `if`(x=0, `if`(y=0
      and k=0, 1, 0), `if`(k2*x*y+x^2-x-y, 0,
      add(b(x-1, y-j, k-(2*y-j)), j=[-1, $1..y])))
    end:
a:= n-> add(b(x, 0, 2*n), x=0..2*n):
seq(a(n), n=0..40);

b[x_, y_, k_] := b[x, y, k] = If[x == 0, If[y == 0 && k == 0, 1, 0], If[k < x || k > 2x y + x^2 - x - y, 0, Sum[b[x - 1, y - j, k - (2y - j)], {j, Join[{-1}, Range[y]]}]]];
a[n_] := Sum[b[x, 0, 2n], {x, 0, 2n}];
a /@ Range[0, 40] (* Jean-François Alcover, Mar 12 2020, after Alois P. Heinz *)

A333017 Twice the total area of all (open or closed) Deutsch paths of length n.

Original entry on oeis.org

0, 1, 6, 25, 90, 306, 1004, 3226, 10218, 32043, 99748, 308787, 951772, 2923563, 8955342, 27368895, 83484042, 254244033, 773219196, 2348780937, 7127522136, 21609615822, 65465845254, 198189732798, 599624708588, 1813169256151, 5480019176754, 16555101318735

Offset: 0

Alois P. Heinz, Mar 05 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2094
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8.
Wikipedia, Counting lattice paths

Cf. A001006, A005043, A330169.

b:= proc(x, y) option remember; `if`(x=0, [1, 0], add((p->
      p+[0, (2*y-j)*p[1]])(b(x-1, y-j)), j=[$1..y, -1]))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;`if`(n<4, [0, 1, 6, 25][n+1],
     ((1045*n^2-4419*n-9646)*a(n-1)-3*(1133*n^2-4679*n-1756)*
      a(n-2)+9*(127*n^2-475*n+480)*a(n-3)+27*(210*n-439)*
       (n-3)*a(n-4))/((n+3)*(83*n-677)))
    end:
seq(a(n), n=0..30);

a = DifferenceRoot[Function[{y, n}, {(-10827 - 16497 n - 5670 n^2) y[n] + (-5508 - 4869 n - 1143 n^2) y[n+1] + (-7032 + 13155 n + 3399 n^2) y[n+2] + (10602 - 3941 n - 1045 n^2) y[n+3] + (7 + n)(-345 + 83 n) y[n+4] == 0, y[0] == 0, y[1] == 1, y[2] == 6, y[3] == 25}]];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 12 2020 *)

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

			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 *)

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A333098 Number of closed Deutsch paths whose area is exactly n.

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A333017 Twice the total area of all (open or closed) Deutsch paths of length n.

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Maple

Mathematica

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.

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Author

Keywords

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Examples

Links

Crossrefs

Programs

Maple

Mathematica