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A333098
Number of closed Deutsch paths whose area is exactly n.
3
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
0,4
COMMENTS
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.
LINKS
Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
MAPLE
b:= proc(x, y, k) option remember; `if`(x=0, `if`(y=0
and k=0, 1, 0), `if`(k<x or k>2*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);
MATHEMATICA
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 *)
CROSSREFS
Cf. A330169.
Sequence in context: A017993 A049870 A093970 * A107428 A086379 A096460
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 07 2020
STATUS
approved