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

0, 1, 6, 25, 90, 306, 1004, 3226, 10218, 32043, 99748, 308787, 951772, 2923563, 8955342, 27368895, 83484042, 254244033, 773219196, 2348780937, 7127522136, 21609615822, 65465845254, 198189732798, 599624708588, 1813169256151, 5480019176754, 16555101318735

Offset: 0

Author

Alois P. Heinz, Mar 05 2020

Keywords

• nonn

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2094
• Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8.
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A001006, A005043, A330169.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, [1, 0], add((p->
      p+[0, (2*y-j)*p[1]])(b(x-1, y-j)), j=[$1..y, -1]))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;`if`(n<4, [0, 1, 6, 25][n+1],
     ((1045*n^2-4419*n-9646)*a(n-1)-3*(1133*n^2-4679*n-1756)*
      a(n-2)+9*(127*n^2-475*n+480)*a(n-3)+27*(210*n-439)*
       (n-3)*a(n-4))/((n+3)*(83*n-677)))
    end:
seq(a(n), n=0..30);

Mathematica

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}]];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 12 2020 *)