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

%I A333098 #16 Mar 12 2020 06:48:37
%S A333098 1,1,1,2,4,6,11,21,36,64,117,208,371,669,1197,2141,3844,6888,12336,
%T A333098 22119,39644,71034,127323,228200,408955,732957,1313647,2354298,
%U A333098 4219447,7562249,13553161,24290307,43533784,78022169,139833177,250612596,449153751,804984038
%N A333098 Number of closed Deutsch paths whose area is exactly n.
%C A333098 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 A333098 Alois P. Heinz, <a href="/A333098/b333098.txt">Table of n, a(n) for n = 0..1250</a>
%H A333098 Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, arXiv:2003.01918 [math.CO], 2020
%H A333098 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A333098 b:= proc(x, y, k) option remember; `if`(x=0, `if`(y=0
%p A333098       and k=0, 1, 0), `if`(k<x or k>2*x*y+x^2-x-y, 0,
%p A333098       add(b(x-1, y-j, k-(2*y-j)), j=[-1, $1..y])))
%p A333098     end:
%p A333098 a:= n-> add(b(x, 0, 2*n), x=0..2*n):
%p A333098 seq(a(n), n=0..40);
%t A333098 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 A333098 a[n_] := Sum[b[x, 0, 2n], {x, 0, 2n}];
%t A333098 a /@ Range[0, 40] (* _Jean-François Alcover_, Mar 12 2020, after _Alois P. Heinz_ *)
%Y A333098 Cf. A330169.
%K A333098 nonn
%O A333098 0,4
%A A333098 _Alois P. Heinz_, Mar 07 2020