A333098 - OEIS

%I #16 Mar 12 2020 06:48:37

%S 1,1,1,2,4,6,11,21,36,64,117,208,371,669,1197,2141,3844,6888,12336,

%T 22119,39644,71034,127323,228200,408955,732957,1313647,2354298,

%U 4219447,7562249,13553161,24290307,43533784,78022169,139833177,250612596,449153751,804984038

%N Number of closed Deutsch paths whose area is exactly n.

%C 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.

%H Alois P. Heinz, <a href="/A333098/b333098.txt">Table of n, a(n) for n = 0..1250</a>

%H Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, arXiv:2003.01918 [math.CO], 2020

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%p b:= proc(x, y, k) option remember; `if`(x=0, `if`(y=0

%p       and k=0, 1, 0), `if`(k<x or k>2*x*y+x^2-x-y, 0,

%p       add(b(x-1, y-j, k-(2*y-j)), j=[-1, $1..y])))

%p     end:

%p a:= n-> add(b(x, 0, 2*n), x=0..2*n):

%p seq(a(n), n=0..40);

%t 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]]}]]];

%t a[n_] := Sum[b[x, 0, 2n], {x, 0, 2n}];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, Mar 12 2020, after _Alois P. Heinz_ *)

%Y Cf. A330169.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Mar 07 2020