A333114 - OEIS

%I #16 Mar 19 2020 10:22:14

%S 1,0,1,1,5,11,44,134,529,1902,7793,31068,133641,574259,2594969,

%T 11842726,56083004,269450143,1333170844,6703500545,34548749471,

%U 181026885253,969167994094,5273977173249,29257773480987,164894374634333,945779302210358,5507572390808676

%N 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 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="/A333114/b333114.txt">Table of n, a(n) for n = 0..800</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>

%e a(4) = (1/1)*(3/1) + 2/2 + 3/3 = 5.

%p b:= proc(x, y, t) option remember; `if`(x=0, 1, add(

%p       `if`(t and j<0, x/y, 1)*b(x-1, y+j, is(j>0)), j=[

%p       `if`(y=0, [][], -1), $1..x-1-y]))

%p     end:

%p a:= n-> b(n, 0, false):

%p seq(a(n), n=0..30);

%t 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 a[n_] := b[n, 0, False];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Mar 19 2020, after _Alois P. Heinz_ *)

%Y Cf. A005043, A005411, A330169, A333098.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Mar 07 2020