cp's OEIS Frontend

A333017 Twice the total area of all (open or closed) Deutsch paths of length n.

%I A333017 #14 Mar 12 2020 06:48:07
%S A333017 0,1,6,25,90,306,1004,3226,10218,32043,99748,308787,951772,2923563,
%T A333017 8955342,27368895,83484042,254244033,773219196,2348780937,7127522136,
%U A333017 21609615822,65465845254,198189732798,599624708588,1813169256151,5480019176754,16555101318735
%N A333017 Twice the total area of all (open or closed) Deutsch paths of length n.
%C A333017 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 A333017 Alois P. Heinz, <a href="/A333017/b333017.txt">Table of n, a(n) for n = 0..2094</a>
%H A333017 Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, arXiv:2003.01918 [math.CO], 2020. See p. 8.
%H A333017 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p A333017 b:= proc(x, y) option remember; `if`(x=0, [1, 0], add((p->
%p A333017       p+[0, (2*y-j)*p[1]])(b(x-1, y-j)), j=[$1..y, -1]))
%p A333017     end:
%p A333017 a:= n-> b(n, 0)[2]:
%p A333017 seq(a(n), n=0..30);
%p A333017 # second Maple program:
%p A333017 a:= proc(n) option remember;`if`(n<4, [0, 1, 6, 25][n+1],
%p A333017      ((1045*n^2-4419*n-9646)*a(n-1)-3*(1133*n^2-4679*n-1756)*
%p A333017       a(n-2)+9*(127*n^2-475*n+480)*a(n-3)+27*(210*n-439)*
%p A333017        (n-3)*a(n-4))/((n+3)*(83*n-677)))
%p A333017     end:
%p A333017 seq(a(n), n=0..30);
%t A333017 a = DifferenceRoot[Function[{y, n}, {(-10827 - 16497 n - 5670 n^2) y[n] + (-5508 - 4869 n - 1143 n^2) y[n+1] + (-7032 + 13155 n + 3399 n^2) y[n+2] + (10602 - 3941 n - 1045 n^2) y[n+3] + (7 + n)(-345 + 83 n) y[n+4] == 0, y[0] == 0, y[1] == 1, y[2] == 6, y[3] == 25}]];
%t A333017 a /@ Range[0, 30] (* _Jean-François Alcover_, Mar 12 2020 *)
%Y A333017 Cf. A001006, A005043, A330169.
%K A333017 nonn
%O A333017 0,3
%A A333017 _Alois P. Heinz_, Mar 05 2020