A330169 a(n) is the total area of all closed Deutsch paths of length n.

1, 3, 12, 39, 129, 411, 1300, 4065, 12633, 39046, 120204, 368844, 1128837, 3447303, 10508592, 31985085, 97226733, 295214316, 895502520, 2714106318, 8219809425, 24877611798, 75248738292, 227488953354, 687408882709, 2076269682831, 6268788729240, 18920387069731, 57086882549253

Offset: 2

Author

Michel Marcus, Mar 05 2020

Keywords

• nonn

Comments

Deutsch paths are a variation of Dyck paths that allow for down-steps of arbitrary length.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..2096
• Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. See p. 8.

Crossrefs

Cf. A001006 (Motzkin numbers), A005043, A333017, A333098.

Programs

Maple

a:= proc(n) option remember;`if`(n<4, [0$2, 1, 3][n+1], (4*n*
      a(n-1)+(2*n+4)*a(n-2)+12*(1-n)*a(n-3)+9*(1-n)*a(n-4))/(n+1))
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Mar 05 2020

Mathematica

a = DifferenceRoot[Function[{y, n}, {9(n+3)y[n] + 12(n+3)y[n+1] - 2(n+6)y[n+2] - 4(n+4)y[n+3] + (n+5)y[n+4] == 0, y[2] == 1, y[3] == 3, y[4] == 12, y[5] == 39}]];
a /@ Range[2, 30] (* Jean-François Alcover, Mar 12 2020 *)

PARI

my(z='z+O('z^30), v=(1-z-sqrt(1-2*z-3*z^2))/(2*z)); Vec(v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2))

Formula

G.f.: v^2*(1+v+v^2)^2/((1+v)^3*(1-v)^2) where v=(1-z-sqrt(1-2*z-3*z^2))/(2*z), that is, where v is the g.f. of A001006.