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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.

%I A333114 #16 Mar 19 2020 10:22:14
%S A333114 1,0,1,1,5,11,44,134,529,1902,7793,31068,133641,574259,2594969,
%T A333114 11842726,56083004,269450143,1333170844,6703500545,34548749471,
%U A333114 181026885253,969167994094,5273977173249,29257773480987,164894374634333,945779302210358,5507572390808676
%N 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.
%C A333114 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 A333114 Alois P. Heinz, <a href="/A333114/b333114.txt">Table of n, a(n) for n = 0..800</a>
%H A333114 Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, arXiv:2003.01918 [math.CO], 2020
%H A333114 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A333114 a(4) = (1/1)*(3/1) + 2/2 + 3/3 = 5.
%p A333114 b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
%p A333114       `if`(t and j<0, x/y, 1)*b(x-1, y+j, is(j>0)), j=[
%p A333114       `if`(y=0, [][], -1), $1..x-1-y]))
%p A333114     end:
%p A333114 a:= n-> b(n, 0, false):
%p A333114 seq(a(n), n=0..30);
%t A333114 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]]}]];
%t A333114 a[n_] := b[n, 0, False];
%t A333114 a /@ Range[0, 30] (* _Jean-François Alcover_, Mar 19 2020, after _Alois P. Heinz_ *)
%Y A333114 Cf. A005043, A005411, A330169, A333098.
%K A333114 nonn
%O A333114 0,5
%A A333114 _Alois P. Heinz_, Mar 07 2020