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Twice the total area of all (open or closed) Deutsch paths of length n.
2

%I #14 Mar 12 2020 06:48:07

%S 0,1,6,25,90,306,1004,3226,10218,32043,99748,308787,951772,2923563,

%T 8955342,27368895,83484042,254244033,773219196,2348780937,7127522136,

%U 21609615822,65465845254,198189732798,599624708588,1813169256151,5480019176754,16555101318735

%N Twice the total area of all (open or closed) Deutsch paths of length 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="/A333017/b333017.txt">Table of n, a(n) for n = 0..2094</a>

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

%p b:= proc(x, y) option remember; `if`(x=0, [1, 0], add((p->

%p p+[0, (2*y-j)*p[1]])(b(x-1, y-j)), j=[$1..y, -1]))

%p end:

%p a:= n-> b(n, 0)[2]:

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

%p # second Maple program:

%p a:= proc(n) option remember;`if`(n<4, [0, 1, 6, 25][n+1],

%p ((1045*n^2-4419*n-9646)*a(n-1)-3*(1133*n^2-4679*n-1756)*

%p a(n-2)+9*(127*n^2-475*n+480)*a(n-3)+27*(210*n-439)*

%p (n-3)*a(n-4))/((n+3)*(83*n-677)))

%p end:

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

%t 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 a /@ Range[0, 30] (* _Jean-François Alcover_, Mar 12 2020 *)

%Y Cf. A001006, A005043, A330169.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 05 2020