%I #59 Dec 19 2024 06:17:24

%S 1,3,12,39,129,411,1300,4065,12633,39046,120204,368844,1128837,

%T 3447303,10508592,31985085,97226733,295214316,895502520,2714106318,

%U 8219809425,24877611798,75248738292,227488953354,687408882709,2076269682831,6268788729240,18920387069731,57086882549253

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

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

%H Alois P. Heinz, <a href="/A330169/b330169.txt">Table of n, a(n) for n = 2..2096</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.

%F 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.

%p a:= proc(n) option remember;`if`(n<4, [0$2, 1, 3][n+1], (4*n*

%p a(n-1)+(2*n+4)*a(n-2)+12*(1-n)*a(n-3)+9*(1-n)*a(n-4))/(n+1))

%p end:

%p seq(a(n), n=2..30); # _Alois P. Heinz_, Mar 05 2020

%t 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}]];

%t a /@ Range[2, 30] (* _Jean-François Alcover_, Mar 12 2020 *)

%o (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))

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

%K nonn

%O 2,2

%A _Michel Marcus_, Mar 05 2020